Strongly Stabilizable Distributed Parameter Systems
This page intentionally left blank
F R O N T I E R S IN APPLIED MATHEMATICS The SIAM series on Frontiers in Applied Mathematics publishes monographs dealing with creative work in a substantive field involving applied mathematics or scientific computation. All works focus on emerging or rapidly developing research areas that report on new techniques to solve mainstream problems in science or engineering. The goal of the series is to promote, through short, inexpensive, expertly written monographs, cutting edge research poised to have a substantial impact on the solutions of problems that advance science and technology. The volumes encompass a broad spectrum of topics important to the applied mathematical areas of education, government, and industry.
EDITORIAL BOARD H.T. Banks, Editor-in-Chief, North Carolina State University Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB Carlos Castillo Chavez, Cornell University Doina Cioranescu, Universite Pierre et Marie Curie (Paris VI) Pat Hagan, NumeriX, New York Matthias Heinkenschloss, Rice University Belinda King,Virginia Polytechnic Institute and State University Jeffrey Sachs, Merck Research Laboratories, Merck and Co., Inc. Ralph Smith, North Carolina State University AnnaTsao, Institute for Defense Analyses, Center for Computing Sciences
BOOKS PUBLISHED IN FRONTIERS IN APPLIED MATHEMATICS
Oostveen, Job, Strongly Stabilizable Distributed Parameter Systems Griewank, Andreas, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation Kelley, C.T., Iterative Methods for Optimization Greenbaum.Anne, Iterative Methods for Solving Linear Systems Kelley, C.T., Iterative Methods for Linear and Nonlinear Equations Bank, Randolph E., PLTMGrA Software Package for Solving Elliptic Partial Differential Equations. Users'Guide 7.0 More, Jorge J. and Wright, Stephen J., Optimization Software Guide Rude, Ulrich, Mothemot/ca/ and Computational Techniques for Multilevel Adaptive Methods Cook, L. Pamela, Transonic Aerodynamics: Problems in Asymptotic Theory Banks, H.T., Control and Estimation in Distributed Parameter Systems Van Loan, Charles, Computational Frameworks for the Fast Fourier Transform Van Huffel,Sabine and Vandewallejoos, TheTotal Least Squares Problem:Computational Aspects and Analysis Castillo, Jose E., Mathematical Aspects of Numerical Grid Generation Bank, R. E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. Users'Guide 6.0 McCormick, Stephen F., Multilevel Adaptive Methods for Partial Differential Equations Grossman, Robert, Symbolic Computation: Applications to Scientific Computing Coleman,Thomas F. and Van Loan, Charles, Handbook for Matrix Computations McCormick, Stephen F., Multigrid Methods Buckmasterjohn D., The Mathematics of Combustion Ewing, Richard E., The Mothematics of Reservoir Simulation
Strongly Stabilizable Distributed Parameter Systems
Job Oostveen Philips Research Laboratories Eindhoven,The Netherlands
Society for Industrial and Applied Mathematics Philadelphia
Copyright © 2000 by the Society for Industrial and Applied Mathematics 109876543 2 I All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688.
Library of Congress Cataloging-in-Publication Data Oostveen.Job. Strongly stabilizable distributed parameter systems / Job Oostveen. p. cm.— (Frontiers in applied mathematics ;20) Includes bibliographical references and index. ISBN 0-89871-455-9 I. Distributed parameter systems. 2. Stability. I.Title. II. Series.
QA402 .058 2000 O03'.78-dc2l 00-026286
is a registered trademark.
Contents Preface
ix
Notation
xi
1 Introduction 1.1 Motivation 1.2 Overview of the Results
1 1 5
2 Two Main Themes 2.1 Strong Stability 2.2 Dissipative Systems with Collocated Actuators and Sensors
9 9 15
3 Strong Stabilizability and Strong Detectability 3.1 Definitions and Motivation 3.2 Equivalence of Internal and External Stability 3.3 The Standard LQ Riccati Equation
23 23 24 26
4
Riccati Equations for Statically Stabilizable Systems 4.1 Problem Formulation 4.2 Mathematical Preliminaries 4.3 Riccati Equations for Stable Systems 4.4 Riccati Equations for Unstable Systems 4.5 Special Types of Riccati Equations 4.5.1 J-Spectral Factorizations 4.5.2 The Strict Positive Real Lemma 4.5.3 The Strict Bounded Real Lemma 4.5.4 The Coercive LQ Case 4.6 Examples 4.6.1 The Strict Positive Real Lemma 4.6.2 The Standard LQ Riccati Equation
31 31 33 39 42 44 44 46 46 47 49 49 49
5
Numerical Approximation 51 5.1 The LQ Riccati Equation for Strongly Stabilizable Systems 51 5.2 Approximation of Strongly Stable Systems 55 5.3 Approximation of the Strongly Stabilizing Solution 60 5.4 Applications to Dissipative Systems with Collocated Actuators and Sensors 62 vii
vlii
Contents
6
Coprime Factorizations and Dynamic Stabilization 6.1 Problem Formulation 6.2 Coprime Factorizations 6.2.1 State-Space Formulas for Normalized Coprime Factorizations . . 6.3 Dynamic Stabilization
67 67 69 72 77
7
Robust Stabilization 7.1 The Nehari Problem 7.2 Parameterization of Robustly Stabilizing Controllers
85 85 96
S
Robustness with Respect to Nonlinear Perturbations 8.1 Preliminaries 8.2 The Popov Criterion 8.2.1 Proof of the Popov Criterion 8.2.2 Asymptotic Tracking
105 106 107 108 112
9 Examples of Strongly Stabilizable Systems 9.1 Boundary Control of a Flexible Beam 9.2 Boundary Control of a Flexible Plate 9.3 Propagation of Sound in a Duct 9.4 A Flexible Structure with Rigid Body Dynamics
115 115 117 120 129
10 Conclusions 10.1 Possibilities for Further Research
137 138
Bibliography
141
Index
149
Preface The theory of distributed parameter systems is becoming more and more a complete theory, containing all of the topics that are well known from finite-dimensional systems theory. However, almost without exception, the theory assumes that the systems are either exponentially stable or exponentially stabilizable. This is rather unfortunate, because there are many models of physically relevant systems that are not exponentially stabilizable. Still, some of these models can be made asymptotically stable in some sense. Most importantly, this is the case for models of weakly damped flexible structures and of acoustical problems. In the literature, there exist many articles containing stability analysis and stabilization methods for such engineering systems. However, unlike the case of exponentially stabilizable systems, until recently there was not a well-structured theory of how to deal with such systems. To be able to deal with such models, ad-hoc techniques had to be applied, very often based on knowledge of a particular Lyapunov function. Moreover, the convergence properties of the approximation methods used were unknown. In this book, we try to fill thi s gap and to provide a framework for a structured approach to strongly stabilizable systems. Most importantly, we present extensions of system theoretic notions and results to the case of strong stability. We believe that this framework will be of use to people working with strongly stabilizable systems, as they will become less dependent on ad-hoc methods. The choice of topics in this book is strongly influenced by the book An Introduction to Infinite-Dimensional Linear Systems by Curtain and Zwart [36]. Moreover, in the same way as [36], our book uses the state-space and frequency domain approach in an integrated fashion. It represents the current status of research for strongly stabilizable systems. We discuss possible definitions of strong stability, stabilizability, and detectability. Moreover, we develop a Riccati theory including numerical approximation of strongly stabilizing solutions and we discuss state-space formulas for (normalized) coprime factorizations. We discuss possibilities of input-output stabilization by dynamic compensators. The robustness of stability is studied, with respect both to nonlinear perturbations and to coprime factor perturbations. We have tried to motivate the results, as much as possible, by supplying a number of models of physical/engineering systems that satisfy the assumptions of our theory. All the theory in this book is restricted to bounded input and output operators. Although this excludes many important examples, there are still many motivating physical systems that do satisfy the assumptions of our theory. These include certain examples of boundary control systems. This book is intended for people working on the analysis and control of systems described by weakly damped partial differential equations, for instance, related to the stabilization of weakly damped flexible structures or to active noise control. Also, it will be of interest to system theorists with an interest in infinite-dimensional systems.
ix
x
Preface
This book is a slightly adapted and extended version of my Ph.D. thesis. As such it represents the outcome of a period of four years at the Department of Mathematics of the University of Groningen, the Netherlands. My supervisor during this period was Ruth Curtain. All results in this thesis are the product of cooperation with her and guidance by her. I would like to express my deepest gratitude for this. I thank Tom Banks for bringing the manuscript to the attention of SIAM and for his invitation for a very inspiring three month stay at his Center for Research in Scientific Computing at North Carolina State University. I have benefitted much from the questions, comments, and suggestions of a number of colleagues, for which I am very grateful. Frank Callier read the entire manuscript of my thesis and came up with an enormous number of corrections and suggestions for improvements. We worked on the results in Chapter 5 with Kazi Ito, whose persistent questions also had a large impact on Chapter 3. Finally, I am grateful for the pleasant cooperation of Marianne Will and her colleagues at SIAM. Job Oostveen Eindhoven, 1999
Notation Symbol
Page
Meaning
A*, adjoint operator of A complex conjugate of s G~(s) = G(-s)* /i 0/2. concatenation of f\ and /2 at T T norm norm in space Z H 2 - or L2-norm HOC- or Loo-norm inner product inner product in space Z
33 35
7 7
{s € C|Re(5) > 0} {s e C|Re(s) > 0}
D(A) Ft F F HG
Hoo(B) HOC
HOC [HO, l-i H2(B)
Hn(a,b)
3
C)
domain of operator A input-output map extended input-output map family of input-output maps Hankel operator with symbol G Hardy space of bounded, holomorphic functions on Cj with values in B Hoo(-B), where B is not specified quotient field of H^ Hardy space of square integrable functions on C^" with values in B space of real-valued functions on the interval (a, 6), of which the derivatives of order 0,..., n are in L 2 (a, 6) imaginary unit
xi
10,35 11 35 87 8 96 7 116
xii_
Notation
C,(Z\,Z-z) L(Z) L2 (a, b; Z)
Banach space of bounded operators from Z\ to Z^ £(Z,Z) class of Lebesgue measurable Z-valued functions, with/ a 6 ||/(*)l| 2 *<^ class of functions which are in L 2 (0,T;Z) for all T>0 class of bounded Lebesgue measurable Z-valued functions on (a, 6)
7
LoofZ)
L^-jOOjOQ'Z)
1
MA Re(s) r(A) T 7n Tjf
class of matrices with elements in A real part of complex number s spectral radius of operator A C0-semigroup Toeplitz operator with symbol II time-domain Toeplitz operator with symbol II
96
time-domain Hankel operator resolvent set of operator A state-space system state-space system, in the case C is irrelevant input map extended input map family of input maps output map extended output map family of output maps
88
oc
L2 (0,co;Z) LOO (a, 6; Z)
7
7
35 33 33
Greek F p(A) Y,(A,B,C,D} E(^4, B, —) $t * ^4 * *
10 10 10 35 10 10 35
Chapter 1
Introduction 1.1 Motivation Questions about stability arise in almost every control problem. As a consequence, stability is one of the most extensively studied subjects in systems theory, and a great diversity of stability notions has been introduced. Many of the prevailing stability definitions are concerned with autonomous differential equations, which in the linear time-invariant case have the form
When Z = R n , such systems correspond to ordinary differential equations and all solutions are given by z ( t ) = eAtz0. In the case that Z is an infinite-dimensional space (we will consider only the case that Z is a Hilbert space), A is a possibly unbounded operator on Z and solutions are given by z(t) = T(t)zQ, where T(i) is the Co-semigroup generated by A (a Co-semigroup is the generalization of e.M to the case of unbounded operators A). Many linear partial differential equations and delay-differential equations can be formulated in this way as first order equations on appropriately chosen Hilbert spaces. For such autonomous systems, by stability we mean global asymptotic stability of the origin; i.e., z(t) tends to zero as t tends to infinity for all initial conditions ZQ € Z. If a finite-dimensional system is globally asymptotically stable, you can even say more: It is exponentially stable; i.e., there exist constants a > 0 and M > 1 such that for all ZQ,
This formula expresses the fact that for exponentially stable systems, there is a uniform bound a on the rate of decay of all trajectories. There are many possible stability definitions for the infinite-dimensional case. In systems theory, exponential stability is the most desirable type of stability. This means that, as for finite-dimensional systems, there is a uniform rate of decay of solutions: there exist constants a > 0 and M > 1 such that for all zo.
Virtually all results in infinite-dimensional systems theory have used exponential stability. There are, however, partial differential equations that are globally asymptotically stable, but /
2
Chapter 1. Introduction
whose solutions do not have a uniform rate of decay. Even if a uniform bound on the rate of decay exists, it may be very hard to find. Thus, there is a need to have stability definitions that do not assume a uniform rate of decay on the solutions. The following example from Russell [91] is a system whose trajectories all converge to zero, but for which there is no uniform rate of decay. Example 1.1.1 The following model describes the displacement in a weakly damped vibrating string of length 1, with clamped ends. The displacement y(x,t) of the string at position x and time t satisfies
with boundary conditions y(0,t) = y(l,t) = 0. Here, D is a damping operator, which is defined below. For easy reference, we define the operator A by Ay = —d^y/dx2, with domain of definition V(A) = {y e ff2(0,l)| ?/(0) = y ( l ) = 1} (#2(0,1) is the space of functions y € L 2 (0,1) that are twice differentiate almost everywhere on (0,1) and whose first- and second-order derivatives are in L2(0,l)). A has eigenvalues \k = fc27r2, k — 1,2,..., with eigenvectors (j>k(x) = \/2sin(k-nx) forming a complete orthonormal set in L 2 (0,l). D is defined by Dy — e(g,y)g, where s > 0 and
with7fc satisfying 0 < |7fc| < M/x/Ait. For example, jk = l/k would satisfy this constraint. Next, we define the Hilbert space Z = L 2 (0,1) x L 2 (0,1). Introducing
(1.1) can be rewritten as
All solutions are given by z(t) = T(t)z0, where T(t) is the semigroup generated by A on Z. Russell has shown that all solutions converge to zero. Furthermore, he showed that all eigenvalues of A have negative real part and are given by
The corresponding eigenvectors ipk form a Riesz basis of Z. If we take the initial condition ^o = V1*:. the solution is given by z ( t ) = e"ktipk and this converges to zero as t tends to infinity. Because the real part of v^ converges to zero as k tends to infinity, there is no uniform bound on the rate of decay. The system is therefore globally asymptotically stable, but not exponentially stable. O This example shows that for partial differential equations the concept of exponential stability is not always applicable. Systems theory for such models should therefore incorporate forms of stability weaker than exponential stability. One such form of stability is strong stability: a semigroup T(t) is strongly stable if all trajectories converge to zero, i.e.,
Section 1.1. Motivation
3
The semigroup occurring in Example 1.1.1 is an instance of a strongly stable semigroup. If the trajectories only converge weakly to zero, the semigroup is called weakly stable:
In this book we concentrate on developing system theoretic results and control theoretic methods for infinite-dimensional systems in the context of strong stability. In systems theory, one does not study autonomous systems, but systems with inputs and outputs. In the linear time-invariant case, these have the form
Again, A, B, C, and D are matrices in the case of a finite-dimensional system and operators in the case of infinite-dimensional systems. In what follows, we will call such systems "control systems" to distinguish them from the "autonomous systems" that we discussed first. Clearly, the different types of asymptotic stability that were defined above for autonomous systems are of importance for control systems, as well, since they concern the solutions with input u(t) — 0. Stability notions for control systems that deal with solutions for u = 0 are called internal stability. However, there is a different class of important notions of stability for control systems, called external stability. These are concerned with the effects of inputs and the effects on outputs. The most important notion is that of input-output stability: We call a system input-output stable if there exists a finite number 7 > 0 such that all solutions ( u ( t ) , z ( t ) , y ( t ) ) with initial condition ZQ = 0 satisfy
If the semigroup of an infinite-dimensional system is exponentially stable, then the system is automatically also input-output stable. Many results in systems theory make use of this implication. For a strongly stable semigroup it does not hold, unfortunately, and this is the cause of a number of complications in the construction of a systems theory for strongly stabilizable systems. It has become clear that calling a control system strongly stable if its semigroup is strongly stable is not an appropriate definition for arriving at useful system theoretic results. This leads us to incorporate certain external stability properties into the definition of a strongly stable system. It is well known that a finite-dimensional linear system is stable if all the eigenvalues of the system matrix A (or all the roots of the characteristic polynomial) are in the left half-plane. Thus, stability can be tested by algebraic methods. For the strong stability of a semigroup, such a simple characterization in terms of spectral properties of its generator A does not exist. Therefore, different, analytical tools must be used to test whether a system is strongly stable. As for nonlinear systems, a Lyapunov analysis can be effective. In the case of physical systems, there is often an obvious choice for a candidate Lyapunov function: the energy of the system. This energy-based approach leads us into the theory of passive systems. Passivity is a notion originating from network theory. Let us first give an example. Example 1.1.2 Consider the model for the motion of a mass-spring-damper system. The displacement of the mass from the equilibrium position is denoted x(t) and an external force u(t) acts on the mass:
4
Chapter 1. Introduction
We take z(t) = ( x ( t ) , x ( t ) ) as the state of this system. The total energy in the system is E(z(t)) = ifcx(f) 2 + ^mx(t)2, and an easy calculation shows that the change of energy satisfies
This inequality expresses the fact that the increase of energy stored in the system does not exceed the work applied by the external force (i.e., it does not exceed the energy supplied by the external source). O Passivity is an abstract version of this physical principle. A control system is called passive if there exists a function 5 : Z —t R + (the storage function) such that the dissipation inequality
is satisfied. As is well known, passivity of linear systems is closely related to the system theoretic property of positive real transfer functions. In this book we will pay special attention to passive/positive real systems, and in many instances we can use the special structure of these systems to obtain sharper results. The initial motivation for the research described in this book was to develop a systems theory for a class of systems occurring as models of flexible structures. These are often modelled as linear systems on a Hilbert space. When the norm is chosen such that it corresponds to the energy of the system, the passive structure is especially highlighted. In this case, they have a state-space representation T^(A,B,B*), where A generates a contraction semigroup. The systems we are interested in here are dissipative systems with collocated sensors and actuators. The term "dissipative" occurs because the generator A of a contraction semigroup is a dissipative operator. For a long time, it has been known by engineers that a partial differential equation describing a mechanical system, such as a flexible structure, leads to a positive real system if actuators and sensors are designed in a "collocated" fashion. This means that measurements and control action are made dual in some sense; a necessary condition for this is that actuators and sensors are implemented at the same location. If an actuator exerts a force at a particular location, a sensor is collocated with this actuator if it measures the linear velocity at the same point. Example 1.1.2 is precisely such an example with collocated actuator and sensor. This is made more precise in Example 2.2.5. Similarly, if the actuator exerts a moment, the sensor must measure the angular velocity at the same location. Having designed collocated actuators and sensors for a dissipative mechanical structure, and thus having obtained a positive real system, negative feedback (i.e., u = -y) stabilizes the system in an input-output sense. This knowledge was used to control flexible structures and the method works even if no detailed model of the system is available, as long as the passive structure is present. In this book, we use the term "dissipative" for operators A that satisfy the relation (Az,z) + (z,Az) < 0 for all z 6 'D(A) (see Definition 2.2.1). To avoid confusion, we would like to stress that we do not use "dissipativity" in its system theoretic or physical sense. In systems theory, dissipativity is a generalization of passivity to allow for different supply rates (the supply rate is the quadratic form in u and y occurring under the integral sign in the dissipation inequality). In this book, we are concerned only with the standard
Section 1.2. Overview of the Results
5
supply rate (u(t),y(t)) and thus we use passivity instead of dissipativity. In physics, the term dissipative is used for systems in which energy is dissipated. For linear systems on an appropriately chosen Hilbert space, this implies the dissipativity of the operator A. However, a dissipative operator A can also represent a system in which all energy is conserved. If in what follows we use the terminology "dissipative system," then we refer to systems in our special class £(.4, B, B*): dissipative systems with collocated actuators and sensors. During the development of this theory, it appeared that many of the results could be formulated for general strongly stabilizable systems, without the assumption of the special structure of dissipative systems with collocated actuators and sensors. Even though we have chosen to give these more general formulations, the models with the special dissipative structure are still the main motivation for this research. Most of our physical examples in Chapter 9 exhibit this form and for many of our results we have given the (often easier) formulation for dissipative systems with collocated actuators and sensors as a corollary.
1.2 Overview of the Results The results in this book can be divided into two parts. First, a framework that is appropriate for studying control systems in the context of strong stability is developed. Second, a diversity of results is obtained within this framework. In this section, we give an overview of the results so that the interested reader can choose his or her favorite route through them.
Chapter 2: Two Main Themes This chapter describes the two central objects of our study. First of all, strong stability, stabilizability and detectability are defined and we derive some important related properties of strongly stable systems. Second, we introduce the class of dissipative systems with collocated actuators and sensors in all its mathematical detail and show the useful properties that it exhibits.
Chapter 3: Stabilizability and Detectability The definitions of strong stabilizability and strong detectability were introduced in Chapter 2 without any motivation. In Chapter 3, we show that these definitions are indeed appropriate by showing that they can be used to generalize certain well-known system theoretic results to the case of strong stability. These results are • the equivalence of strong stability and input-output stability for systems that are strongly stabilizable and strongly detectable; • the existence of a unique stabilizing solution to the LQ Riccati equation for a strongly stabilizable and strongly detectable system. These results have also been published in [28].
Chapter 4: Riccati Theory Probably no control theoretic tool has been studied more thoroughly than the Riccati equation. Therefore, a logical topic with which to start the development of systems theory for
6
Chapter I. Introduction
strongly stabilizable systems is the theory of existence and uniqueness of strongly stabilizing solutions to algebraic Riccati equations. A start at this subject was made in Chapter 3 with results for the standard LQ Riccati equation. In Chapter 4, we study very general Riccati equations for strongly stabilizable systems. The results were published in [80]. It is shown here that a strongly stabilizing solution exists for a Riccati equation with a general (i.e., not necessarily positive) cost operator and associated with a statically stabilizable system if and only if a certain factorization of the Popov function exists. This basic result is used to derive more specific results for both general statically stabilizable systems and dissipative infinite-dimensional systems with collocated measurement and control. This includes a result for the existence of J-spectral factorizations and versions of the positive real lemma and the bounded real lemma.
Chapter 5: Numerical Approximation If the theory of Chapter 4 is to be used in a practical control design, then one needs to be able to compute numerical approximations to the operator solution of the Riccati equation. It is important to use approximation algorithms with proven convergence properties. In Chapter 5, the convergence of sequences of stabilizing solutions of finite-dimensional algebraic Riccati equations to the strongly stabilizing solution of the standard LQ Riccati equation is studied and proved. This result was obtained in collaboration with Curtain and Ito [83]. In Chapter 9, we illustrate the effectiveness of the algorithm with a numerical example of the LQ Riccati equation for a problem of sound propagation.
Chapter 6: Coprime Factorizations and Compensators In the foregoing chapters, stabilization by state feedback or by static output feedback has been considered. Of course, it is also important to have available a theory about dynamic compensation. The relation between dynamic stabilization and coprime factorizations is well known, so in this chapter the two are treated jointly. We give state-space formulas for normalized doubly coprime factorizations for a class of strongly stabilizable and detectable systems. Furthermore, we develop results about dynamic (input-output) stabilization of strongly stabilizable systems. We also study whether the resulting closed-loop system is strongly stable. Unlike the results in the rest of this book, we do not obtain a perfect generalization of the finite-dimensional results for the subjects in this chapter. The chapter ends with a positive result about strong stabilization of a system in the ~E(A,B,B*) class by finite-dimensional strictly positive real controllers. Results about normalized coprime factorizations for systems in the "E(A,B,B*) class were published in Oostveen and Curtain [81]. The generalization to more general strongly stabilizable and detectable systems appeared in Curtain and Oostveen [29].
Chapter 7: Robust Stabilization The question of robustness of control systems has been very prominent in control theory in the 1980s and 1990s. For infinite-dimensional systems it is of even greater importance, because LQG and H^-controllers are usually also infinite-dimensional. So, for implementation purposes, one will have to approximate these controllers. In this chapter, we give a parameterization of controllers that robustly stabilize a strongly stabilizable plant with respect to coprime factor perturbations. An important step in the solution of this problem is the explicit parameterization of all solutions to a certain Nehari extension problem.
Section 1.2. Overview of the Results
7
The results on the Nehari problem were published in [27]. The papers [30, 81] concern the parameterization of robustly stabilizing controllers for the £(^4, B, B*) class and more general strongly stabilizable and detectable systems, respectively.
Chapter 8: Nonlinear Perturbations In Chapter 8, we deal with a classical robustness problem: the absolute stability problem. A version of the Popov criterion is presented as a solution. We show that a strongly stable system with strictly positive real transfer function remains asymptotically stable under a class of nonlinear perturbations in the feedback loop. This result is published in [82] and a preliminary version appeared in [26].
Chapter 9: Examples of Strongly Stabilizable Systems Four examples of partial differential equation models of engineering systems are described in Chapter 9. We show how they are cast into state-space form, in which three of them exhibit the special dissipative structure; so, in particular, they are examples of strongly stabilizable systems. For one of the examples we work out the numerical approximation procedure for the strongly stabilizing solution of the standard LQ Riccati equation, which was described in Chapter 5.
Chapter 10: Conclusions In this last chapter, we summarize the research described in this book and we discuss some open questions and possibilities for future research in the area of strongly stabilizable infinite-dimensional systems.
Notation The relevant notation is summarized in a list on pages xi and xii. All notation is defined at the location where it is first used. As an exception to this rule, we give here definitions of the function spaces used for systems and signals. Let Z be a Hilbert space and B a Banach space. Then L 2 (a,6;Z) is the space of weakly measurable, Z-valued functions on (a,b), such that Il/ll2 = (/ a b ||/(t)ll!^<°o; La(Z) is the space of weakly measurable, Z-valued functions defined on the imaginary axis, such that ||/||2 = (/^ ||/(jo;)|||dw)i < co; L 2 oc (0,oo;Z) is the Frechet space of weakly measurable, Z-valued functions that are in L 2 (0,r ; Z)forallT>0; LOO (a, b', B) is the space of weakly measurable, 5-valued, essentially bounded functions, defined on (a, 6); LOO (5) is the space of weakly measurable, B-valued, essentially bounded functions, defined on the imaginary axis, with norm ||/||oo = ess sup^gn ||/0'w)||oo; H-2(Z) is the space of holomorphic, Z-valued functions defined on the open right halfplane, such that ||/||2 = supI>0(/^o \\f(x + jy)\\2zdy)? < oo;
8
Chapter I. Introduction
H^-(Z) is the left half-plane counterpart of H 2 (Z). Hoo(B) is the space of holomorphic, J5-valued functions defined on the open right-half plane, such that \\f\\oo =supx>0\\f(x+jy)\\z < oo. Note that || • ||oo denotes the norm both in Lc>0 and HOQ. This does not cause any problems, because if a function / is in HQQ, then it has an extension to the imaginary axis and its Hoc-norm equals the Loo-norm of its boundary function f(ju). A similar remark applies to the norms of LQ and H2. In the above definitions, all types of L2- and Hg-spaces are Hilbert spaces. In Chapter 7, we also use H^B), with B a finite-dimensional Banach space. It is defined in the same way as H 2 (Z), above. However, because B has no inner product, Ht(B) has no inner product either, and it is not a Hilbert space.
Chapter 2
Two Main Themes This book is centered around two main themes. The first one is strong stability. As we have explained in the introduction, it is not always possible to work with the notion of exponential stability. Therefore, we define the weaker notion of strong stability for control systems in Section 2.1. In Section 2.2, we consider the second main theme: the class E(A,B,B*) of dissipative systems with collocated actuators and sensors. This class contains many physically motivated examples of strongly stabilizable systems. This second theme is directly linked to the notion of passivity. Systems in the Y,(A,B,B*) class are passive, and this property is exploited frequently to obtain useful results.
2.1 Strong Stability In this book, we develop system and control theoretic methods for nonexponentially stabilizable systems. In the system theoretic literature on infinite-dimensional linear systems, the prevailing concept has always been exponential stability. In the literature on partial differential equations, however, it is customary to consider merely global asymptotic stability of the trajectories. As a second step, one might try to obtain a uniform rate of decay for the trajectories (for this reason, exponential stability is often called uniform stability in these papers). For many partial differential equations (for instance, undamped wave equations and beam equations with certain boundary conditions), exponential stability is not obtained, even though global asymptotic stability is. The above considerations are concerned only with stability of the state. As we explained in Chapter 1, a system with an exponentially stable semigroup also satisfies certain external stability properties, and these properties are important to arrive at useful system theoretic results. Unfortunately, systems that exhibit only asymptotic convergence of the state to zero do not satisfy these external stability properties, in general. Therefore, an appropriate stability definition should include the behavior of inputs and outputs. To this end, we will introduce the notion of a strongly stable system in Definition 2.1.1 and related stabilizability notions in Definition 2.1.7. A discussion of a number of aspects of the stability definition completes this section. An extensive treatment of the suitability of our definitions of strong stabilizability and strong detectability follows in Chapter 3. Our definitions apply to linear time-invariant control systems of the form
9
10
Chapter 2. Two Main Themes
where z(t) € Z, u(t) € U, y(t) e Y and U,Y,Z are separable Hilbertspaces. Furthermore, A generates a Co-semigroup of bounded operators T(t), B € £.(U,Z), C € L(Z,Y) and D 6 £(U,Y). We denote this system by H(A,B,C,D). If the value of D is not of interest, or if D = 0, then we denote the system by £(A,B,C). To define strong stability for such a system we have to introduce a number of maps. First, we define the input map $ t :L 2 (0,oo; [/)->£ by
For all t > 0, there exists a positive number kt such that
Next, we define 3>t : L 2 (0, oc; t/) -> Z by
Defining the reflection operator Rt: L2(0, oo; U) —» L2(0, oo; U) by
we obtain ||-Rtu||i, 2 (o,t;E/) = IH|z,2(<w/) and $t = *t-Rt- Obviously, this implies that ||$t|| = ||$t|| < fc(. Under the assumption that the operators $« are uniformly bounded, i.e., there exists a A; > 0 such that for all u
we can take the limit for t —> oo of $4:
and $ € £(L 2 (0, oo; U), Z). This limiting operator $ is called the extended input map. Second, let us introduce the output map tyt : Z -> L 2 (0,t;y):
Again, these operators are bounded and they can be extended to an operator taking values inL 2 oc (0,oo;F):
^ is called the extended output map. If the operators $t are uniformly bounded, then
$e£(z,L 2 (o,oo ; r)).
Finally, we define the input-output map W t : L2(0,co;{7) —> L2(0,t;F) by
Section 2.1. Strong Stability
11
If the operators F( are uniformly bounded, then
exists, where F € £(L2(0,oo;C/),L 2 (0,oo;F)), and is called the extended input-output map. The operators above stem from the definition of well-posed linear systems. This very general class of linear infinite-dimensional systems, which allows for very unbounded input and output operators, was introduced by Weiss (see [106, 107, 108] or the survey by Curtain [24]). In the case of an exponentially stable semigroup T(t), the maps $t, Vt and Ft are always uniformly bounded, and so the extended output map \t, the extended input map $ and the extended input-output map F all exist and are bounded operators. If T(t) is only strongly stable, then in general this is not the case. This fact motivates the following definition of a strongly stable system. Definition 2.1.1 Consider the system Y,(A,B,C,D) and let T(t),
z€Z.
• H(A,B,C,D) is input-output stable if and only if its transfer function G(s) = D + CV-^-'BeHoo^C/.y)). It can be easily seen that if A generates an exponentially stable semigroup T(t), E(A,B,C,D) is input stable, output stable and input-output stable. Hence, every exponentially stable system is a strongly stable system. The intuition behind the definition of strong stability is that a system is strongly stable if, with any initial condition and any input in 1^(0, oo; [/), the state is a bounded function of time and the output is in L 2 (0,oo;y) and, furthermore, that the state converges to zero whenever the input function equals zero. In fact, it can also be proved that for nonzero
12
Chapter 2. Two Main Themes
u e L2(0,oo;C7), the state converges to zero. This is done in the following lemma, which will turn out to be of crucial importance when obtaining system and control theoretic results for strongly stable systems. Lemma 2.1.3 Assume that the system E(A,B,C, D) is input stable and that A generates a strongly stable semigroup. Then, for any u € L,2(0, co; U), the solution z(i) of (2.1) converges to zero ast^oo. Proof The solution z(i) is given by
The first term tends to zero because T(t) is a strongly stable semigroup. We rewrite the second term in the following way:
Now, since T(t) is strongly stable and u(t) e L 2 (0, oo; U), we have for fixed ti.
For the remaining term, we have
where we define
and <3>t and $t are defined as in (2.2), (2.3). It is easy to see that \\$t\\ = \\$t\\ < ||$||. Hence,
By the input stability, ||$ | < oo, so the above expression can be made arbitrarily small by choosing ti large enough, which completes our proof. D The notions of input stability and output stability were also used in Hansen and Weiss [53], but the terminology used there is different. If the system is input stable, then Hansen and Weiss call B an infinite-time admissible control operator for T(t). Similarly, our notion of output stability is equivalent to C being an infinite-time admissible observation operator forT(t). In the same paper, Hansen and Weiss show that there is a strong relation between input stability and the solvability of a particular Lyapunov equation. Lemma 2.1.4 (Hansen and Weiss [53, Theorem 3.1]) System E(A, B, C, D) is input stable if and only if there exists at least one self-adjoint, nonnegative solution II £ £-(Z) to the Lyapunov equation
Section 2.1. Strong Stability
13
Moreover, in this case, the operator P £ £-(Z) defined by
is the smallest nonnegative solution of (2.9). By duality, similar statements hold relating output stability to the solvability of the dual Lyapunov equation A*Q + QA = —C*C. In addition to the result in Lemma 2.1.4, Hansen and Weiss showed that input stability of E(j4, B, C, D) has a number of implications for the system, in particular for the semigroups T(t)andT*(t). Lemma 2.1.5 Assume that E(A,B,C, D) is input stable and let P be given by (2.10). 1. ForaUz£Z,\\mt^x>P?T*(t)z = Q. 2. ff P is boundedly invertible, then T* (r.) is a strongly stable semigroup. 3. // P > 0 and T(t) is uniformly bounded, then T*(t) (and hence also T(t)) is a weakly stable semigroup, i.e., limt->.00{T*(t)zi,Z2) — 0 for all zi,Z2 € Z. 4. If T* (t) is a strongly stable semigroup, then P is the unique self-adjoint solution of (2.9). Again, dual statements hold for output stable systems. Remark 2.1.6 Note that for an input stable system, approximate controllability is equivalent to ker($*) = {0} (see Theorem 4.1.22 in Curtain and Zwart [36] for this equivalence). Therefore, in this case, Y,(A,B,C) is approximately controllable if and only if P>0. In view of the third result in the above lemma, it is interesting to mention the following result (see Remark 3.3 in Hansen and Weiss): Let A be the generator of a weakly stable semigroup T(t). Both T(i) and T*(t) are strongly stable semigroups if • A has compact resolvent (i.e., for some (and hence for any) s e p(A), the resolvent operator (si — A)~[ is compact) or
• cr(A)njHt is at most countable. The last condition is a consequence of an important result by Arendt and Batty [2]. They show that if T(t) is uniformly bounded, cr(A) njR is countable and A has no eigenvalues on the imaginary axis, then T(t) is a strongly stable semigroup. Note that their result is applicable, because a weakly stable semigroup is uniformly bounded and the generator of a weakly stable semigroup cannot have purely imaginary eigenvalues. The result by Arendt and Batty is one instance of a wealth of results about the relation between strong stability of a semigroup and the spectrum of its generator. Unlike the case of exponentially stable semigroups, a necessary and sufficient condition on the spectrum of the infinitesimal generator A for strong stability of the semigroup T(t) generated by A is not known. There are, however, a number of necessary conditions and a number of sufficient conditions known. Huang [54], for instance, has shown that if T(t) is uniformly bounded for t > 0 and cr(A] is contained in the open left half-plane, then T(t) is strongly
14
Chapter 2. Two Main Themes
stable. In the same paper he also proved a necessary condition: If T(t) is strongly stable, then it is uniformly bounded and a(A) is contained in the closed left half-plane. Batty [15] showed that, in this case, there cannot be eigenvalues on the imaginary axis. An important special case of the result by Arendt and Batty, in view of the class of systems in Section 2.2, is the following: If A generates a contraction semigroup, has compact resolvent and has no eigenvalues on the imaginary axis, then it generates a strongly stable semigroup. Now that we have a definition of strongly stable systems, we also need related notions of stabilizability and detectability. These are introduced next. In Chapter 3, we will elaborate on why these notions of stabilizability and detectability are appropriate. Definition 2.1.7 • The system ~S(A,B,C,D) is strongly stabilizable if there exists an F £ £(Z,U) such that E(A + BF,B: [F* C* } * , D) is output stable and A + BF generates a strongly stable semigroup. • The system H(A,B,C, D) is strongly detectable if there exists an L e £(Y, Z) such that T,(A + LC, [L B } , C, D) is input stable and A + LC generates a strongly stable semigroup. • The system Y,(A, B, C, D) is statically stabilizable if there exists a K € £(Y} U) such that Z(A + BKC,B,C, D) is a strongly stable system. In [ 101], Staffans also introduced concepts of strong stabilizability and strong detectability. Even though our definition of a strongly stable system is the same as the one by Staffans [97], our definitions of strong stabilizability and strong detectability are quite different from his, and in Chapter 3 we will discuss the difference. Our definition of static stabilizability is new. It was introduced in Oostveen and Curtain [80], where it was used fruitfully to construct a Riccati theory for strongly stabilizable systems. Moreover, in the following section we identify a class of models that are statically stabilizable. Our definitions of strongly stable systems (Definition 2.1.1) and of stabilizability and detectability (Definition 2.1.7) are independent of the feedthrough operator D of the system. This is also the case for the definition of static stabilizability. Therefore, it corresponds only to the stabilization by the static output feedback law u = Ky in the special case that D = 0. If D ^= 0, then the static output feedback u = (I + KD)~lKy would lead to the strongly stable closed-loop system Z(A + BKC,B, (I + DK)C,D). In the 1970s and 1980s there has been considerable interest in weak and strong stabilizability of semigroups. Most of these results concern contraction semigroups, or semigroups which are in some sense similar to a contraction. A number of different approaches were used to obtain these results. The earliest papers were written by Slemrod [92, 93]. The paper [92] concerns weak stabilizability of contraction semigroups. In [93], Slemrod used a Lyapunov approach to show that if A is skew adjoint and generates a contraction semigroup, (A, B)is approximately controllable and the trajectories satisfy a certain compactness condition, then the feedback u = —B*z renders the closed-loop semigroup strongly stable. Another approach was based on a decomposition of the state space in the subspace on which the contraction semigroup is unitary and its orthogonal complement, called the completely nonunitary part of the state space. Many stabilizability results were based on this decomposition, for instance, Benchimol [ 18,19], Levan [62,67] andLevan andRigby [68]. Benchimol [19] derived a necessary and sufficient condition for strong stabilizability. He showed that if A generates a contraction semigroup and has compact resolvent, then the semigroup generated by A — BB* is strongly stable if and only if the "weakly unstable
Section 2.2. Dissipative Systems with Collocated Actuators and Sensors
15
states" of T(t) are approximately controllable. The relation between the decomposition approach and the Lyapunov-based approach was studied in Levan [63]. Later, a Riccati equation approach was used to prove the stability of the semigroups generated by A — BB* and A - BB*P, where P is a nonnegative solution of the standard LQ Riccati equation. Among these papers are Balakrishnan [4] and Levan [64, 65,66]. A spectral approach was taken by Batty and Phong [16]. They showed that if A generates a contraction semigroup and cr(A) n j'R is countable, then the semigroup generated by A — BB* is strongly stable if and only if there do not exist purely imaginary eigenvalues of A with a corresponding eigenvector in ker(j9*). The special class of dissipative systems with collocated actuators and sensors, which will be introduced in the next section, consists of systems whose semigroups satisfy the assumptions about strong stabilization of semigroups of most of the above papers, in particular those in Benchimol [19]. In this book, we are interested in the stronger form of stabilizability of the whole system, as in Definition 2.1.7. New results about these notions of stabilizability and detectability are presented in Chapter 3. It is well known that if Y,(A,B,C) is exponentially stabilizable and B is a bounded, finite-rank operator, then A can have only finitely many eigenvalues in the closed right half-plane (see, for instance, Theorem 5.2.3 in Curtain and Zwart [36]). We would like to have similar information for the case of a strongly stabilizable system E(A,B,C, D) with bounded, finite-rank B. We can obtain this by applying the theorem about the exponentially stabilizable case. This theorem states that if the system T,(A,B,C,D) is /3-exponentially stabilizable, then there exists a constant 5 < (3 such that S(j4) n {s e C | Re(s) > 5} consists of a finite number of eigenvalues with finite multiplicities. A system is called /?exponentially stabilizable if there exists F € C(U, Y) such that eatTBf(t) is exponentially stable for some a > —/?, where TBF(£) is the semigroup generated by A -f BF. We can apply this theorem with arbitrary /? > 0, because if TBF(^) is a strongly stable semigroup, then eatTBF(t) is exponentially stable for all a < 0. So, according to the theorem, for all /? > 0 there does exist a 5 < (3 such that there are only finitely many points with real part greater than 5 in the spectrum of A (counting multiplicities) and all of these spectral points are eigenvalues. Consequently, A has only countably many eigenvalues in the open right half-plane, and in that case these eigenvalues have an accumulation point, which must lie on the imaginary axis. Moreover, there is no continuous or residual spectrum of A in the open right half-plane. If the right half-plane eigenvalues of A do not have an accumulation point on the imaginary axis, then there are only finitely many of them.
2.2 Dissipative Systems with Collocated Actuators and Sensors By far the most important class of systems that are strongly stabilizable and detectable is the class of dissipative systems with collocated actuators and sensors, which we will often denote as the Y,(A,B,B*) class. Although mathematically very special, they have a prominent place in the engineering literature, mainly as models for flexible structures. The most important mathematical characteristics are that A generates a contraction semigroup and C = B*. The duality condition C — B* can arise if actuators and sensors are implemented at the same location; hence the terminology "collocation." The term dissipative comes from the fact that the generator of a contraction semigroup is a dissipative operator. These systems are often found in the engineering literature as models in acoustics and for
16
Chapter 2. Two Main Themes
flexible structures (see, for instance, the monograph by Joshi [57], which deals with (finitedimensional) models of flexible structures). Some simple physical examples of dissipative systems with collocated actuators and sensors are the models for transverse vibrations of a flexible beam with boundary control in Slemrod [94], for vibrations in a flexible plate in You [ 120] and for the propagation of sound in a tube terminated by linear oscillators in Ito and Propst [56]. At the end of this section, we give the model from Slemrod. The other two models are given in Chapter 9. An important motivation for studying systems in this class was the benchmark problem SCOLE (Spacecraft Control Laboratory Experiment), which was posed by NASA for the control of flexible spacecraft. This model, which was published in Balakrishnan [5], describes an antenna on a flexible mast which is supported by the space shuttle. This benchmark problem has generated much research in control of flexible structures described by partial differential equations. The models in the "^(A^B^B*) class are the classical example of systems that are not exponentially stabilizable. The spectrum of A can have infinitely many points on the imaginary axis or arbitrarily close to it. A case in which this happens is, for instance, when A is skew-adjoint: (Az,z) + (z,Az} = 0 for all z € TJ>(A). In this case, all eigenvalues of A are on the imaginary axis. It follows from Theorem 5.2.3 in Curtain and Zwart [36] that if there are infinitely many points of the spectrum of A arbitrarily close to the imaginary axis, then (A, B) cannot be exponentially stabilizable if B is compact, for instance, if B is bounded and has finite rank. In this book, we only consider systems with a bounded input operator B. Note that this does not exclude examples of boundary control. In fact, the first three examples in Chapter 9 are controlled on the boundary. Even though these systems are not exponentially stabilizable, they do have many useful properties, most importantly regarding stabilizability and robustness. We will give the precise conditions under which we study these systems and then list some important structural properties, many of which will be used intensively throughout this book. First we need to introduce three concepts which are important for our class of systems. Definition 2.2.1 Let Abe a possibly unbounded operator on a Hilbert space Z, with domain of definition V(A) C Z. A is a dissipative operator if
Definition 2.2.2 A linear system Y,(A,B,C,D) is passive if there exists a function S : Z -> R + (called the storage function) such that the dissipation inequality
is satisfied for all inputs u(t) and all initial conditions z(0) € Z. The concept of a passive linear system is closely related to the notion of a positive real transfer function (see Willems [117, 118] for this relation). Definition 2.2.3 Let U and Y be real Hilbert spaces. An £(U,Y)-valued transfer function G(s) is positive real if
Section 2.2. Dissipative Systems with Collocated Actuators and Sensors
17
When the third inequality is replaced by G(s)* 4- G(s) > el for some e > 0, then we call G(s) strictly positive real. We need to explain what we mean by G(s), as it is not a priori clear what is meant by the complex conjugate of an operator between two infinite-dimensional spaces. In the finite-dimensional case, we usually have U = Rm and Y = Rp, and because s can b complex, we have to consider the complexification of Y. The same phenomenon occurs in the infinite-dimensional case. U and Y are real Hilbert spaces, and if s is complex, then for u&U, G(s)u 6 Y + jY. Thus, G(s) can be written as G(s) = Gr(s) +jGc(s), where both Gr and Gc take values as operators between the real spaces U and Y. Now, clearly G(s) can be defined as G(s) = Gr(s) —jGc(s) and the first assumption in Definition 2.2.3 can be reformulated as Gr(s) = G r (s), Gc(s) — -Gc(s). We consider systems given by
where z£.Z,u&U,y£U and Z, U are real separable Hilbert spaces. Furthermore, we assume the following: Al. A generates a Co-semigroup of contractions T(t) on Z. A2. Be£(U,Z). A3. ^(A, —,B*) is approximately observable, A4. S(j4, B, —) is approximately controllable. AS. A has compact resolvent. Let us briefly discuss these assumptions. Often, the Hilbert space Z is equipped with a norm which corresponds to the total energy of the system. In that case, assumption Al expresses the fact that the energy does not increase if the input is identically equal to zero. The Lumer-Phillips Theorem (see Pazy [85, Section 1.4]) states that if A generates a contraction semigroup T(i), then it is dissipative (see Definition 2.2.1). Assumption A5 is satisfied for most partial differential equations on compact domains. It is well known that if a closed operator A has compact resolvent, then its spectrum consists only of eigenvalues of finite multiplicity (see Section III.6.8 in Kato [59]). In the literature on the control of flexible structures, one often finds systems of the Y,(A,B,B*) structure, satisfying assumptions A1-A5. The following beam example has been taken from Slemrod [94], who attributes it to Bailey and Hubbard [3]. Example 2.2.4 Let us consider a flexible cantilever beam of length L. A EulerBernoulli model is used to model the transverse vibrations w(x, t) of the beam. A mass is attached to the tip of the beam and a piezoelectric film is bonded to one side of the beam. When a voltage is applied to the film, it will apply a bending moment to the beam. This voltage is the control input for the system. We measure the angular velocity of the tip. The following model is suitable for this system:
18
Chapter 2. Two Main Themes
with boundary conditions
and measurement
Slemrod [94] has shown that the system (2.14)-{2.18) can be represented by the abstract differential equation on a certain Hilbert space Z,
with a bounded input operator B. The norm on the Hilbert space Z corresponds to the energy in the system. In the same paper, he showed that assumptions A1-A5 are satisfied for this model. Hence, the model has a realization in the Y,(A:B,B*) class. In Chapter 9 we discuss his state-space representation of this model in more detail. <> In Chapter 9, we present three more examples of partial differential equations that fit into the Y,(A,B,B*) framework. It is remarkable that the £(A,B,B*) structure, which occurs so frequently in the case of infinite-dimensional systems, is hardly ever encountered in passive finite-dimensional systems. The reason for this is that in case of finite-dimensional systems, one generally uses the standard inner product of R". When modelling infinitedimensional systems it is very common to use an inner product corresponding to the energy in the system. It is only in this energy-related inner product that the system exhibits its special E(A,B,B*) structure. Let us illustrate this remark with an example of a massspring-damper system. Example 2.2.5 We consider a mass-spring-damper system, with an external force u(t) acting on the (unit) mass:
The velocity of the mass is observed:
If we choose z = co\(x,x) as the state vector on the state space R2 with the usual inner product, then we obtain a state-space representation
where
Section 2.2. Dissipative Systems with Collocated Actuators and Sensors
19
In this case, we do not have A + A* < 0. However, we could introduce as the state space R2 with the different inner product
In this case,
i.e., the sum of the potential and kinetic energy in the system. On this state space, we still have C — B*, but now,
which leads to
Thus, we have found a realization satisfying the T,(A, B1B") structure, by carefully choosing the inner product on the state space. O Systems in the *£(A,B,B*) class satisfy a number of useful properties. The most important ones for this book are proven in the following lemma. Lemma 2.2.6 Let Z, U and Y be real separable Hilbert spaces and let AB := A — BB*. PL If assumptions Al andA2 hold, then AB generates a Co-semigroup of contractions Tg(t). Under the additional assumptions A4 and A5, this semigroup is strongly stable. P2. If assumptions A J andA2 hold, then A*B generates a Co-semigroup of contractions TB ((). Under the additional assumptions A3 and A5t this semigroup is strongly stable. P3. Under the assumptions Al andA2, tl(A,B,B*) and'S(AB,B,B*) are passive systems and have positive real transfer functions. P4. Under the assumptions Al andA2, H(AB,B,B*) is output stable and \\B*(SI-AB)^z\\*
P6. Under the assumptions Al andA2, S(j4j3, £?,£*) is input-output stable and
HB^SI-AB^B^^I.
Proof PI. By the Lumer-Phillips Theorem, assumption Al implies that A is a dissipative operator. Now, for z € T>(A),
20
Chapter 2. Two Main Themes
Therefore, AB is also dissipative. Since \\T(t)\\ - \\T*(t)\\, T*(t) is also a contraction semigroup. Again applying the Lumer-Phillips Theorem, it follows that A* is dissipative. It now follows in a similar way to that above thai: A*B is also dissipative. Thus, both AB and A*B are dissipative. Applying Corollary 4.4 from Pazy [85], it follows that Tg(i) is a contraction semigroup. TB (t) was proven to be a strongly stable semigroup in Corollary 3.1 of Benchimol [19], under assumptions Al, A2, A4 and A5. P2. The proof of P2 is completely analogous to the proof of PI, this time using the assumptions Al, A2, A3 and A5. Note that if A generates a contraction semigroup, then so does A*, and if A has compact resolvent, then so has A*. P3. Assume that assumptions A1 andA2hold. Letz0 € ~D(A) and let u be differentiable almost everywhere on [0,fi) with derivative ii(t) € Li(0,ii;t/). Then the function z ( t ) on [0, ti) given by z(t) = T(t)z0 + J0* T(t - s)Bu(s)ds is the strong solution of (2.13). In particular, it is differentiable for almost all t e [0,ti). Now, using the dissipativity of A,
Integrating this inequality from 0 to t\ and defining the storage function S(z) = | ||z obtain the dissipation inequality
This inequality is extended in the usual way to the case of arbitrary z0 € Z and u e 1^2(0,00; C7). Hence, the system Z(A,B,B") is passive. To prove the positive real property, note that for all u € U and for s e CQ~ ,
The first inequality follows from the dissipativity of A. The dissipativity also implies that CQ~ is contained in the resolvent set of A and so B* (si - A)~1B is holomorphic in the open right half-plane. Thus, under the assumptions A1 and A2, B*(sl — A)~1B is positive real. Because E(.AB,£?,.£?*) also satisfies assumptions Al and A2, it follows that it is passive and has a positive real transfer function as well. P4. For P4, we need to show that B*(sI-AB)~lz e H 2 (£7) for all z € Z, or, equivalently, by the Paley-Wiener Theorem (see, for instance, Theorem A.6.21 in Curtain and Zwart [36]), that
Section 2.2. Dissipative Systems with Collocated Actuators and Sensors
21
Now, by assumption Al, A is dissipative. Hence, for all z € "D(A),
Integrating this inequality, we obtain
Since T>(A) is dense in S, this inequality extends to all z € Z. Taking the limit for t —>• oo then leads to
P5. P5 is proved analogously to P4, by differentiating ||TJ3(t),j||2 instead. P6. For the input-output stability, we refer to a result from Curtain and van Keulen [33]. It states that if P(s) is positive real, then (7 + P(s))~ 1 6 HOC- Hence, with P(s) = B*(sl - A)~1B, which is positive real by P3,
Next, we prove that \\B*(sI - AB)~1B\\00 < 1. Denoting G(s) = B*(sl - AB)~1B, we deduce for Re(s) > 0 that
where the inequality follows from the dissipativity of A. Thus,
So, G"(s)G(s) < I for all Re(s) > 0, which implies that HG^ < 1. D Summarizing, the above lemma shows that if A generates a contraction semigroup and B is bounded, then 51(AB,B,B*) is input stable, output stable and input-output stable. Under the additional assumption that the pair (A,B) is approximately controllable and A has compact resolvent, S(ylB,B,S*) is a strongly stable system. Note that the observability assumption A3 was not necessary for this. It was used only to prove the strong stability of the adjoint semigroup Tg(t). In fact, the set of assumptions Al, A2, A4 and A5 can be used to prove the strong stability of both semigroups Ts(t) and Tg(t), using part 3 of Lemma 2.1.5 and Remark 2.1.6. Similarly, the strong stability of both semigroups can be proved to follow from the set of assumptions Al, A2, A3 and A5.
22
Chapter 2. Two Main Themes
The fact that, under the assumptions A1, A2, A4 and A5, the statement that E ( AB ,B,B*) is a strongly stable system can be rephrased by saying that the feedback u = —y is strongly stabilizing for E(A,B,B*) (the system is statically stabilizable with K = —I). It is easy to show that exactly the same holds for the feedback u(t) = —ky(t) for any k > 0. The combination of P3 with P6 is a nice instance of the famous passivity theorem. This classical theorem states that the negative feedback interconnection of a positive real system (in our case T,(A,B,B*)) with a strictly positive real system (here, the constant gain /) is input-output stable. This particular property is the most important reason why collocation of actuators and sensors is used so often in the engineering literature on control of flexible structures. Also, it is known that these collocated systems have good robustness properties. For instance, in Curtain and van Keulen [33] it was shown that infinite-dimensional positive real systems can be robustly stabilized with respect to coprime factor perturbations, with a robustness margin of at least l/V%- In fact, a controller which achieves this robustness margin is u(t) = -y(t). We can also deduce information on the location of zeros for single-input-single-output systems in the E(A, B, B*) class. Let us consider G(s) =d + B*(sI-A)~1B with d > 0. From the proof of Lemma 2.2.6, it follows that
So, if A is a zero of G, i.e., G(A) = 0, then Re(X)\\(XI - A)~l B\\ < -d. I f d > 0 , then this implies that Re(A) < 0, i.e., G is minimum phase. For the case d = 0, let us take an arbitrary A with Re(A) > 0. As A is dissipative, A £ p(A). So, (A/ - A)~1B = 0 implies that B = 0. This contradicts the controllability assumption A3. Thus, in the case d = 0, G(A) = 0 can hold only for Re(A) < 0. We conclude that, if d = 0, then G(s) has only zeros in the closed left half-plane, and for d > 0 they are in the open left half-plane.
Chapter 3
Strong Stabilizability and Strong Detectability In the previous chapter, we have set the scene for the remaining chapters. In particular, we have defined the notion of a strongly stable system in Definition 2.1.1. Of course, after defining stability, one should think about suitable definitions of Stabilizability and detectability. There are many different candidate definitions for strong Stabilizability and strong detectability and the ones that we gave in Definition 2.1.7 may not be the most intuitive ones. Indeed, other definitions may be suitable in certain circumstances. In this chapter, we will elaborate on the definitions of our choice.
3.1 Definitions and Motivation In Definition 2.1.7, we introduced the following Stabilizability and detectability definitions. • The system E(A, B, C) is strongly stabilizable if there exists an F e £(Z, U) such that ^(A + BF, B,[F* C" ] *) is output stable and A + BF generates a strongly stable semigroup. • The system E(A,B, C) is strongly detectable if there exists an L € £(Y,Z) such that E(A + LC, [ L B ] , C ) is input stable and A + LC generates a strongly stable semigroup. • The system ^(A,B,C) is statically stabilizable if there exists a K 6 £(U,Y) such that E(A + BKC, B, C) is a strongly stable system. Note that, for notational convenience, we often write
The motivation for our definitions of strong Stabilizability and strong detectability is twofold. First, our definitions allow us to extend a number of well-known system theoretic results to the case of strongly stabilizable systems. These results are the equivalence of internal and external stability for systems that are stabilizable and detectable and the existence of a unique stabilizing solution to the LQ Riccati equation for a stabilizable and 23
24
Chapter 3. Strong Stabilizability and Strong Detectability
detectable system. Second, we show that if a system is strongly stabilizable and strongly detectable, then there exists a feedback operator FQ for which Y,(A + BFo,B, [ F£ C* ] ) is a strongly stable system. In the following section, we deal with the equivalence between internal and external stability. In Section 3.3, we prove the existence and uniqueness of a strongly stabilizing solution to the LQ Riccati equation and we show that the existence of an operator F0, as above, is a direct consequence of the results on Riccati equations. We remark that, contrary to the case of exponential stability, our notions of strong Stabilizability and strong detectability are not completely dual. The missing element for duality is that if A + LC generates a strongly stable semigroup, then it does not automatically follow that A* + C* L* generates a strongly stable semigroup. This lack of duality is directly due to the nature of strong convergence and, thus, it is intrinsic to every sensible definition of strong Stabilizability. Before we go on to the next section, it is interesting to compare our definition with the definition of strong Stabilizability in Staffans [98]. Although his definition of a strongly stable system is exactly the same as ours, his definition of strong Stabilizability is different. After a little reformulation, his definition of strong Stabilizability states that there must exist F e £(Z, U) such that E(A + BF, B,[C* F* ] *. £>) is a strongly stable system. The difference from our definition is obvious: Staffans requires input-output stability and input stability of the closed-loop system, which we do not demand. As we will show in the remainder of this chapter, for many results we do not need input-output stability and input stability to be included in the definition of strong Stabilizability.
3.2 Equivalence of Internal and External Stability In this section, we show that our definitions of strong Stabilizability and strong detectability are appropriate to generalize a well-known finite-dimensional result about the equivalence of internal and external stability to the case of strong stability. Let us first formulate this finite-dimensional result. Theorem 3.2.1 Consider the finite-dimensional system S(i4,B,C7). A is a stable matrix if and only if (A, B) is stabilizable, (A, C) is detectable and C(sl — A)~1B € HOQ. The version for the case of a strongly stabilizable, strongly detectable systems reads as follows. Theorem 3.2.2 H(A,B,C) is a strongly stable system if and only if it is strongly stabilizable, strongly detectable and input-output stable. Proof The sufficiency is obvious, taking F = 0 and L — 0. For the necessity, assume that C(sl - A)~lB e HOC (£([/, y)) and that S(A, B, C) is strongly stabilizable and strongly detectable. Hence, there exists an F e £(/?, U) such that C(sl - Ap)~lz e H 2 (F) andF(s/-A F )- 1 z€H 2 (f/),andthere exists anLe£(Y,Z) such \halB*(sI-A*L)-lz£ H 2 (C/) and L*(sl - A*L}~lz e H 2 (y) for all z € Z. Using the perturbation formulas
we obtain that for all z 6 Z the left-hand sides of these two equations are in \{^(Y) and H2(C/), respectively, which proves that Y,(A,B,C) is input stable and output stable. To
Section 3.2. Equivalence of Internal and External Stability
25
prove the strong stability of the semigroup T(t) generated by A, we consider the perturbation formula
Since ~£(A,B,C) is output stable, CT(-)z € L2(0,oo;y) and the strong stability of Ti(t) and the input stability of E(A + LC,L,—) then prove that T(t)z —> 0 as t —> oo, using Lemma 2.1.3. D We remark that if E(^4,S,C") is exponentially stabilizable and detectable in the sense of Definition 5.2.1 of Curtain and Zwart [36], then it is automatically strongly stabilizable and detectable. In [36], they obtained a nice generalization of Theorem 3.2.1 to the case of infinite-dimensional exponentially stabilizable and detectable systems, but the additional assumption that the dimensions of U and Y are finite was needed. Our Theorem 3.2.2 shows that in the case of infinite-dimensional input and output spaces, one does at least obtain strong stability. An analogous result for exponential stability of a large class of systems with unbounded input and output operators is reported by Rebarber [89]. It is easy to see that static stabilizability implies strong stabilizability (taking F — KC) and strong detectability (taking L = BK). Therefore, the following corollary is immediate. Corollary 3.2.3 Y,(A,B,C) is a strongly stable system if and only if it is statically stahilizable and input-output stable. Let us further examine the connection between static stabilizability and the duo of strong stabilizability and strong detectability. As we just mentioned, static stabilizability implies strong stabilizability and strong detectability. Lemma 3.2.4 gives a complete characterization of the relation between the two concepts. Lemma 3.2.4 H,(A,B,C)is statically stabilizable if and only if it is both strongly stabilizable and strongly detectable and there exists a K e £ (Y, U) such that G(I - KG) ~1 e HOO(£([/, Y)), where G(s) = C(sI-Ar1B. Proof In this proof, we will use the shorthand notation AK = A + BKC, AL = A + LC andAF = A + BF. First, note that G(s)(I - KG(s))~l = C(sI-AKrlB- Indeed,
Necessity: If T,(A, B, C) is statically stabilizable, i.e., if there exists K e £(Y, U) such that S(j4x, B, C) is a strongly stable system, then it is also strongly stabilizable and strongly detectable, taking F = KC and L = BK. Furthermore, G(I - KG)~l is the input-outpu map of the strongly stable system £(AK,B,C) and so is in Hoo(£(t/,y)). Sufficiency: Assume that E(j4,j5,C) is strongly stabilizable and strongly detectable and that G(I - KG)~l € H00(£(C7,y)) for some K e C(Y, U). We show that X(A, B, C) is statically stabilized by the same K. G(s)(I — KG(s))~l is the input-output map of %(AK,B,C) and so this system is input-output stable. By the strong stabilizability, there exists F e C.(Z,U) such that E(AF,B, [ F* C* ]*) is output stable. Thus, we have
which shows that C(sl - AK)~IZ € H 2 (Y) for all z € Z, i.e., E(AK,B,C) is output stable. Similarly, the strong detectability implies that there exists an L € £(Y, Z) such that
26
Chapter 3. Strong Stabilizability and Strong Detectability
E(j4i, [B L ] ,C) is input stable. Now,
shows that B*(si- A*K)~lz & H 2 (C/) for all z € Z, i.e., E(AK,B,-) is input stable. It remains to show that the Co-semigroup Tj((t) generated by AK is strongly stable. The detectability assumption also implies that AI, generates a strongly stable semigroup TL(£). Consider the perturbation formula
Because !/,(£) is a strongly stable semigroup, E (AL, [B L ] ,C) is input stable and Y,(AK,B,C) is output stable, we can apply Lemma 2.1.3 to show that Tfc(£) is a strongly stable semigroup. D
3.3 The Standard LQ Riccati Equation In this section, we prove that our definitions of strong Stabilizability and strong detectability allow us to generalize an essential result about the existence of a strongly stabilizing solution to the standard LQ Riccati equation from the finite-dimensional case to the case of strongly slabilizable and detectable infinite-dimensional systems. We consider the standard LQ optimal control problem for a system E(v4,S,C), that is, we want to find an input u € 1^(0, oo; U) for Y,(A, B, C) such that a given quadratic cost criterion is minimized. Let us first formulate the finite-dimensional result. Theorem 3.3.1 Consider the cost criterion
where u(t) is the input of the finite-dimensional system !£(A, B, C) and y(t) is its output. 1. If(A,B)isstabilizable, then there exists a minimizing control u E L2(0, oo;f7), which is given by ii(t) = —B*Xoz(t), where XQ is the minimal self-adjoint, nonnegative solution to the algebraic Riccati equation
The corresponding minimal cost is J(u, ZQ) = (XQZQ,ZQ). 2. If, in addition, (A, C) is detectable, then X0 is the unique self-adjoint, nonnegative solution of (3.2) and A - BB*X0 is stable. For the strongly stabilizable and strongly detectable case, we have the following result. Theorem 3.3.2 Consider the system T^(A, B, C) with cost criterion (3.1). 1. IfE(A,B,C) is strongly stabilizable, then there exists a minimizing control u 6 L2(0,oo; U), which is given by u(t) = -B*Xoz(t), where X0 is the minimal selfadjoint, nonnegative solution to the algebraic Riccati equation
Section 3.3. The Standard LQ Riccati Equation
27
for z e T>(A). The corresponding minimal cost is J(U,ZQ) = (XQZO,ZO}. Moreover, Z(A-BB*X0,B, [ ~(B*X0)* C" ]*) (i.e., the corresponding closed-loop system) is output stable and input-output stable. 2. If, in addition, T,(A, B, C) is strongly detectable, then X$ is the unique self-adjoint, nonnegative solution of (3.4). Moreover, the corresponding closed-loop system is a strongly stable system. Proof 1. Let F be the feedback operator that is strongly stabilizing for T,(A, B, C) and choose u(t) = Fz(t) to obtain
for some c > 0, since £(v4 + BF,B, [F* C* ] *) is output stable. So, the system satisfies the conditions of Theorem 6.2.4 in Curtain and Zwart [36] and there exists a minimal self-adjoint, nonnegative solution X$ of (3.3) such that the minimal cost is »7(u,zo) = (XOZQ,ZO). From this we immediately obtain the following estimates, which we will use in the proof of part 2:
and
where Tx0(t) is the semigroup generated by AXO = A — BB*Xo. These estimates show that Y,(AXo,B, [ -(B*X0)* C* ]*) is output stable. Next, we prove that the closed-loop system is input-output stable. We define M (s) = I-B*X0(sI-AXu)-lB andN(s) = C(sI-AXo)-1B. Note that I- M and N are the two components of the closed-loop transfer function, and so we need to show that M and N are in H^. It follows from the output stability of the closed-loop system that M(s) and N(s) are holomorphic for all s € Cj. Hence, for all s € Cj}~, M(s) and N(s) are well defined and we can deduce the following identity:
So, for all s € CQ" and all u e U,
28
Chapter 3. Strong Stabilizability and Strong Detectability
Thus, both M(s) and N(s) are holomorphic and uniformly bounded in the open right halfplane, i.e., M € Hoo(£(tf)) a^ ^ e H co (£(t/,r)). 2. We first prove the input stability of the closed-loop system, using the perturbation formula
The result follows from the input stability of E(/lz,, [ B L ],-) and the input-output stability of Z(AXo,B, [ (B*X0)* C* ]*). Next, we show that the semigroup TXo (t) is strongly stable. To do this, we write
and so
Now, by the strong stability of TL(*)> ^(*)zo -> 0 as t -» oo. Furthermore, ui(t) = Cz(t) e L 2 (0,oo ; y) by (3.5) and u2(t) = S*X0z(<) € L 2 (0,oo;t/) by (3.4). So, we can apply Lemma 2.1.3 to show that the integral term tends to zero as t tends to infinity. Thus, we obtain that z(t) tends to zero as t goes to infinity, i.e., TXo(t) is strongly stable. Finally, the uniqueness of X0 follows in exactly the same way as in the case of an exponentially stable semigroup (see, for instance, Lemma 6.2.6 and Theorem 6.2.7 in Curtain and Zwart [36]). D Remark 3.3.3 Note that in the proof of the above theorem, the strong stability of the semigroup generated by Ap was not used. Hence, the result of part 1 of this theorem is already valid if there exists an F e £(Z, U) such that E(A + BF,B, [F* C* ] *) is output stable. The strong stability of the semigroup generated by AL was used only to prove the strong stability of the semigroup T\0(t) and the uniqueness of the solution XQ. We also need the dual result of Theorem 3.3.2. Because it is completely analogous we state it without proof. Theorem 3.3.4 1. IfS(A,B,C) is strongly detectable, then there exists a minimal self-adjoint, nonnegative solution Y0 to the algebraic Riccati equation
for z e V(A*). Moreover, S(A - Y0CC*, \ B -F0C* ] ,C) is input stable and input-output stable. 2. If, in addition, there exists an F such that (A + BF)* generates a strongly stable semigroup and "E(A + BF,B, [ F* C* ] *) is output stable, then Y0 is the unique self-adjoint, nonnegative solution of (3.3), (A — Y$CC*)* generates a strongly stable semigroup and H,(A - Y0CC*, [ B -Y0C* ] , C) is input stable, output stable and input-output stable.
Section 3.3. The Standard LQ Riccati Equation
29
Combining Theorem 3.3.2 with Theorem 3.3.4, we are immediately given the following result. Corollary 3.3.5 Suppose that "£(A, B, C) is strongly stabilizable and strongly detectable. Then there exists an L0 € £(Y,Z) such that E(A +L0C,[ L0 B ] * ,C) is input stable, output stable and input-output stable and there exists an FQ e £(Z, U) such that T,(A + BF0,B, [ F0* C* ] *) is a strongly stable system. Proof The result follows immediately from Theorems 3.3.2 and 3.3.4, by taking FQ = -B*X0 and L0 = -Y0C*. D One would like to conclude that H(A + L^C, [ LQ B ] , C) is also a strongly stable system, but A + LoC does not generate a strongly stable semigroup, in general. This is exactly the lack of duality that was discussed at the end of Section 3.1. On the other hand, from Theorem 3.3.2 it follows that (A + L^C)* does generate a strongly stable semigroup. So, in the special case that A has compact resolvent, this implies the strong stability of the semigroup generated by A + LoC. Example 3.3.6 As an example of this lack of duality for strong stability, consider the shift-semigroup T(t) on the Hilbert space
defined by
It is well known that lim^oo T(t)f = 0 for all / € Z, i.e., T is a strongly stable semigroup on Z. However, T*(t) is given by
Clearly, for all f e Z and all t > 0, \\T*(t)f\\z = \\f\\z- Thus, it follows that T*(t) is not strongly stable. Finally, we show what results can still be obtained if we weaken the stabilizability and observability assumptions. Lemma 3.3.7 Consider the system E (A, B, C) and assume that there exist F € £ (Z, U) and L e £(Y, Z) such that
then TBF(^). tne semigroup generated by A + BF, is uniformly bounded if and only if the semigroup generated by A + LC, TLc(t), is also uniformly bounded. Moreover, TBF(£) is a strongly stable semigroup if T^c(t) is strongly stable. Proof Let $BF denote the extended output map of HBF and $>LC denote the extended input map of S^c1. These two maps are bounded because of our assumptions. Now, for any z € Z and t > 0,
30
Chapter 3. Strong Stabilizability and Strong Detectability
By the output stability, we have for all z e Z,
From this we can obtain that
The assertion of strong stability follows directly from Lemma 2.1.3. D This lemma has an important consequence for systems £(.A, B, B*) which only satisfy assumptions Al and A2. In that case, Lemma 2.2.6 tells us that E(^4 — BB*,B,B*) is input stable, output stable and input-output stable and the semigroup TB(£) generated by A - BB* is a contraction semigroup. By looking at the proof of part 2 of Theorem 3.3.2, it follows that there exists a minimal, self-adjoint, nonnegative solution XQ to the Riccati equation
and that the system E(A - BB*XQ,B, [ B (B*X0)* ] *) is output stable. Applying the lemma above, it follows that the semigroup TX (t) generated by A — BB*Xo is uniformly bounded. Moreover, if Tg(f) is a strongly stable semigroup, then T\(t) is also strongly stable. Thus, we have recovered a major part of the results of Theorem 3.3.2 under very weak assumptions. In this chapter we have only considered stabilization via static feedback. Of course, stabilization by dynamic compensators is also very important and will be studied extensively in Chapter 6.
Chapter 4
Riccati Equations for Statically Stabilizable Systems In the previous chapter, we derived a result for the existence of a strongly stabilizing solution of the standard LQ Riccati equation for a strongly Stabilizable and strongly detectable system. It is important to have results available for more general Riccati equations. It will turn out that nice results can be obtained, using the concept of static stabilizability. Therefore, in this chapter we present a theory for the existence of strongly stabilizing solutions to much more general Riccati equations for statically Stabilizable systems.
4.1 Problem Formulation In this chapter, we consider the following Riccati equation related to the infinite-dimensional system E(A,B,C):
Throughout, we assume that N € £(Y, U), Q = Q* € C(Y), R = R* € L(U) and that R is boundedly invertible. Our aim is to derive conditions for the existence of stabilizing solutions to this Riccati equation. We define what we mean by a stabilizing solution in the context of strongly stabilizing systems. Definition 4.1.1 The operator X € £(Z) is a strongly stabilizing solution of the Riccati equation (4.1) if
where AFx =A + BFX and Fx = -R~l(B*X+ NC). The approach we take is a spectral factorization approach. This means that the condition for existence of a strongly stabilizing solution will be related to the existence of a spectral factorization of the Popov function associated with the Riccati equation. Definition 4.1.2 The Popov function associated with the Riccati equation (4.1) is the function O : jR -> C(U) given by
31
32
Chapter 4. Riccati Equations for Statically Stabilizable Systems
where G(jw) = C(juil -A)~1B. We remark that a sufficient condition for the Popov function II to be well defined is that the extended input-output map ofS(A,B,C),W, is a bounded operator from L2(0, oo; U) to L2(0,oo; Y). In this case, G € H 00 (£(f/,F)) and has an extension to the imaginary axis in the sense that lim^o G(a +ju;)u exists for all u€.U and for almost all w € R (see Theorem 4.5 in Rosenblum and Rovnyak [90]). Moreover, in this case, II e L00(£([/)). The type of factorization that we consider is the so-called Wiener-Hopf factorization. Definition 4.1.3 Consider a function II € L a5 (>C(C/)). A Wiener-Hopf factorization of II is a factorization where M, N, M~l, N~l e !!«(£([/)). A sufficient condition for II to have a Wiener-Hopf factorization is that it satisfies the coercivity condition See Theorem 5 in Devinatz and Shinbrot [40] or Rosenblum and Rovnyak [90, Theorem 3.7]. Particular cases of Wiener-Hopf factorizations are • spectral factorizations, where N = M, and
There have been many papers concerning the relation between Riccati equations and spectral factorizations. For the case of finite-dimensional systems, we mention Molinari [74] and Kucera [60]. Callier and Winkin [21] and Weiss [111, 112] deal with the case of exponentially Stabilizable infinite-dimensional systems. More recently, there has been a spate of papers using a spectral factorization approach to solutions of Riccati equations for a class of systems much more general than the class considered here, namely, the class of well-posed linear systems (see Staffans [96, 97, 98, 99], Weiss and Weiss [114] andMikkola [73]). The generality of the class of well-posed linear systems lies in the fact that B and C (and other operators) are allowed to be very unbounded; in our case, they are assumed to be bounded. So, although our class of systems is more restrictive in that B and C are assumed to be bounded, the type of Riccati equation considered is quite general and most importantly we obtain necessary and sufficient conditions for the existence and uniqueness of strongly stabilizing solutions. Our approach follows the thesis of Weiss [111] very closely (see also his paper [112]). An essential difference between our approach and the development of the Popov approach in Weiss [111, 112] lies in the initial formulation of the problem. In these papers, he took C = I and this forced him to work with exponential stability, instead of strong stability. As in Weiss and Weiss [114], we utilize the extra C-factor to weaken the stability requirements. Let us consider the cost functional associated with (4.1),
subject to the constraints
Section 4.2. Mathematical Preliminaries
33
Using the extended output map ^ and the extended input-output map of the system £ (A, B, C), we can write the output y as
As in Weiss [111] and Weiss and Weiss [114], we reformulate the cost as
where
If we assume that E(j4,B,C) is a strongly stable system, then \I> € £(Z, 1,2(0, oo;V)) and F € £(L2(0,oo; t/),L2(0,oo;y)) and this is sufficient to ensure that J is well defined (that is, bounded) for all inputs u 6 1^(0, oo; U). If C = /, then to ensure that $ and F are bounded, one needs to assume that T(t) is exponentially stable, as was done in Weiss [111]. The introduction of the extra C-factor weakens this to the assumptions of output stability and input-output stability. In Section 4.2, we give the background needed to prove our result. This includes some theory of well-posed and regular linear systems and basic results regarding Toeplitz operators and Wiener-Hopf factorizations. In Section 4.3, we prove the existence and uniqueness of a strongly stabilizing solution to the Riccati equation for strongly stable systems. In Section 4.4, we extend our results to the statically stabilizable case. We illustrate the usefulness of these results by applying them to obtain versions of the positive and bounded real lemma and to obtain results about the standard LQ Riccati equation and the existence of J-spectral factorizations. In Section 4.6, we apply the results to systems in the Y>(A,B,B*) class.
4.2 Mathematical Preliminaries In this section we give definitions and results needed to prove the main results. Toeplitz operators play a central role in our theory. Definition 4.2.1 Let II 6 Loo(£(Z)). The Toeplitz operator with symbol II is defined by
where P+ is the orthogonal projection from L2(— joo,joc;Z) onto H^Z). domain Toeplitz operator with symbol U is defined by
The time-
for z 6 L 2 (0, oo; Z), where $ denotes the Fourier transformation that maps L 2 (0, oo; Z) ontoH2(Z). That 7n and 7n are bounded operators follows from Theorem A.6.26 in Curtain and Zwart [36]. We state some properties of Toeplitz operators, which we will use in this chapter. The proofs of the following statements can be found in Section 2.1 of Weiss [111]. For a complex function G(s), we define <3~ by G~(s) := G(-s)*.
34
Chapter 4. Riccati Equations for Statically Stabilizable Systems Lemma 4.2.2 1. lfG~ 6 H 00 (£(Z)) or K € H*, (£(£)), then TGK = TGTK. 2.IfG£ Hoo(jC(Z)), then TG equals the multiplication map induced by G. Consequently, in this case, TG = FG, where WG is the extended input-output map of a system with transfer function G. 3. The adjoint of the Toeplitz operator TG is given by (7c)* = TG~-
The invertibility of the Toeplitz operator which has the Popov function as its symbol is important in the proof of our main result. Lemma 4.2.3 Consider the Popov function H defined by (4.2) and the operator Tt defined by (4.5). Then 1. 71 equals the time-domain Toeplitz operator T^; 2. 7L is boundedly invertible if and only i/7n is,' 3. 7n is boundedly invertible if U admits a
Wiener-Hopffactorization.
Proof 1. This is proven in Lemma4.27 in Weiss [111]. 2. This follows from part 1 and the fact that 7n and T^ are isomorphic. 3. See Theorem 5 of Devinatz and Shinbrot [40]. We shall often use the following result to prove that certain functions are in Hx. The result is well known in harmonic analysis and a proof for the scalar case can be found in Duren [42]. The proof for the operator-valued case is analogous and we give it here for completeness. Lemma 4.2.4 Let F : C|J~ —» jC(C7, Y), where U and Y are separable Hilbert spaces. Assume that F(ju) € L00(£({/,y)). Recall that F~(s) := F(-s)*. 1. IfF(s)u e H2(Y)forallu e U, then F(s) 6 H 00 (£(C7,r)). 2. IfF~(-a)y
€ H2(U)forally£
Y, then F(s) € H00(£(t/,r)).
Proof 1. F(s) is holomorphic for Re(s) > 0, because F(s)u € H 2 (F) for all u € U. Let us denote HFI^ = M and s = a + j/3. By Theorem 11.2 in Duren [42], we have the following Poisson representation of F(s) for Re(s) = a > 0:
So, for every s € Cj,
Hence, F(s) is bounded and holomorphic on CQ , i.e., F(s) e Hoo(£(C/,y)). 2. It can easily be checked that F(s) € Loo(£(l/,y)) holds if and only if F~(-s) € L 00 (£(t/,y)) and that F(s) € H00(£(C/,y)) if and only if F~(-s) £ H00(£([/,y)). Using this, part 2 is a direct consequence of part 1. D
Section 4.2. Mathematical Preliminaries
35
Next, we need some theory about well-posed linear systems and regular linear systems. Details and further references can be found in Weiss [110] or in the survey paper by Curtain [24]. For any Hilbert space Z, ST is the right shift on L2(0, oo; Z), defined by
An operator F on L2 (0, co; Z) is called shift invariant if FST = STW. PT denotes the projection of L 2 (0,oo;Z) onto L 2 (0,T;Z) by truncation, defined for / 6 L 2 (0,oo;Z) by
F°r A,/2 € L2(0, oo; Z) and r > 0, the r-concatenation of /i and /2, denoted /i 0/2, is T defined by
Note that /i 0/2 = Prfi +ST/2- We now define well-posed linear systems. Definition 4.2.5 A well-posed linear system on the Hilbert spaces U, Z and Y is a quadruple (T,$,$,F), where T
1. T = {T(£)}t>o, where T(t) is a strongly continuous semigroup of bounded linear operators on Z; 2. $ = {$t}t>o is a family of bounded linear operators from L,2(0,oo; U) to Z such that
for any u,v € L2(0, oo; U) and any r,t>03. 4? = {^t}t>o is a family of bounded linear operators from Z to L,2(0,oo;y) such that
for any z € Z and any T, < > 0 and \J>0 = 0;
4. F = {Ft}t>o is a family of bounded linear operators from L2(0, oo; U) to L 2 (0,oo;y) such that
for any u, v € L2(0, oo; U) and any r,t>0 and FQ = 0. The four families of operators defined above relate the input function u € L 2 (0,t; U), the output function y € L 2 (0,t; Y"), the state at time t, z ( t ) € Z and the initial state ZQ e Z of the system on any finite interval [0,£) in the following way:
36
Chapter 4. Riccati Equations for Statically Stabilizable Systems
It is not a coincidence that we used the same notation for the defining maps of well-posed linear systems and the maps in Section 2.1. With every well-posed linear system we can associate a triple of operators (.A, B, C), where B and C are (possibly unbounded) operators from U to Z and Z to Y, respectively. These operators are such that A is the generator of T(t), $t satisfies (2.2) and *
• oo along the real axis. A sufficient condition for a well-posed linear system to be regular is that the input operator B be bounded. A regular linear system is generated by a quadruple (A,B,C,D) of operators such that (2.2) and (2.5) hold and, furthermore,
where Du = lims-xx, G(s)u and CL is an extension of C; to a domain which includes T(t)z for every z e Z and almost every t (CL is called the Lebesgue extension of C; see Weiss [107]). If C is bounded, then CL = C and the formula for F reduces to (2.7). (A, B,C,D) are called the generating operators of the system. Now, we are back in the situation that we described in Chapter 2, with the only exception being that B and C can be unbounded operators. Hence, under the assumption of uniform boundedness of <E>t, ^ and Ft, we can define the extended input map <£, the extended output map Vf and the extended input-output map F for such a system. Note that in the bounded case, the system is always regular and has the transfer function G(s) = D + C(sl — A)~1B. In our applications, B and C will always be bounded but we have to be careful about the boundedness of the input and output operators of realizations of spectral factors. If we wish to emphasize that we consider a system with bounded input and output operators, we use the terminology "bounded linear system." We also need some results about dual systems lor bounded linear systems. The dual system Sd = S(A*,C*,B*,£>*) can be defined in an obvious way with A* the generator of the Co-semigroup T* (•) on Z. It follows directly from the frequency domain criteria for input stability and output stability in Lemma 2.1.2 that Y,(A,B,C) is output stable if and only ifS(A*,C*,B*) is input stable. Moreover, for any y 6 L2(0,oo;y) with derivative y € L 2 (0,oo;y) there holds
and
for all t > 0. For more general duality results for well-posed linear systems see Section 6 of Weiss and Weiss [114]. The following result was proven in Theorem 11.1 of the same paper (see also Lemma 4.10 of Staffans [101]). It is not particularly obvious, even for bounded linear systems; therefore, we include a proof for this special case.
Section 4.2. Mathematical Preliminaries
37
Lemma 4.2.6 Let T(-) be a Co-semigroup and let \t be an extended output map forT(-). 1. IfV e £(L 2 (0,oo;f7),L 2 (0,oo;y)) is shift invariant, then Vnew = F*# is an extended output map for T(-). 2. Let (T,t>,4',F) represent a bounded linear system with the generating operators (A, B, C,D) and suppose that Z(A,B,C,D) is output stable. Then, ^new = F** = CnewT(-) is an extended output map for T(-), where
Proof 1. We need to show that tynew satisfies part 3 of Definition 4.2.5, which reduces to showing that
This follows from
2. From (2.5) it follows that
Taking now ZQ <= T>(A), we see that w = ^fz0 and w = ^Az0 are both in L 2 (0,oo;y) by assumption. Thus (4.6) and (4.8) hold, yielding
Since all the operators on the right-hand side arc bounded, Cnew is in C(Z,Y). The given expression for V = ^!*^ follows from (4.6). D Finally, we conclude this section with another technical lemma that was proven in Theorem 11.3 in Weiss and Weiss [114]. For completeness, we give a proof for the special case of a bounded linear system. Lemma 4.2.7 Let ^(A,B,C) be a strongly stable bounded linear system and let the operators N, Q and R be given. Suppose that the corresponding Popov function II e LOO(£([/)) has a Wiener-Hopffactorization, U(JLJ) = M~(ju)N(jw). Then N ( - ) is the transfer function of a well-posed linear system (T, $, *jy, FN), where the Co-semigroup and the input map are the same as those for E(A, B, C). Proof Let II have a Wiener-Hopf factorization II = M~N. This implies that M is invertible over HQQ (£(£/)). We denote the extended input-output map corresponding to the transfer function M by FA/- Because FA/ and M are isomorphic via the Fourier transform (see Theorem A.6.27 in Curtain and Zwart [36]), F^/ is bounded and shift invariant.
38
Chapter 4. Riccati Equations for Statically Stabilizable Systems Next, we define
Because F and FM are shift invariant and bounded and Q and TV are bounded, it follows that F is a shift invariant and bounded operator from L2(0, oo; U) to L2(0,oo; Y). So, from Lemma 4.2.6, we can define a new output map \&/v for T(t) by
Next, we have to show that F^ satisfies the functional equation in part 4 of Definition 4.2.5:
We can split the above equation into two equations, one for 0 < t < T and one for t > T. The first one is trivially satisfied using the causality of ¥N; the second one becomes
If we apply a shift over a distance r to this equation, we see that we must check whether Fjv satisfies
where ST is the right shift on L 2 (0,oo;t/). Now, since (T,$,*,F) is a well-posed linear system,
and so
From the shift invariance of F, we obtain
Next,
By Lemma 4.2.2, 7^ = 7^/~7^ = F^Fjv, which in combination with the above shows that
and since FM is invertible we obtain
Section 4.3. Riccati Equations for Stable Systems
39
Using a similar argument as that for F one can show that the right-hand side is invariant with respect to the left shift 5* and so the same holds for the left-hand side. Therefore,
and so
This shows that (T,$,*^,FAT) is a well-posed linear system with the transfer function
N(S).
a
In the general theory in Weiss and Weiss [114], neither N nor M needs to be regular. However, in the case of a bounded linear system, N is always regular, since it has the same bounded input operator B as the system.
4.3 Riccati Equations for Stable Systems In this section we will prove our main result of this chapter, which relates strongly stabilizing solutions of the Riccati equation to Wiener-Hopf factorizations. However, before doing so, we prove that the Riccati equation has at most one strongly stabilizing solution. The proof is completely analogous to the one of Lemma 3 in Curtain and Zwart [37]. Lemma 4.3.1 If the Riccati equation (4.1) has a self-adjoint solution X which is such that A + BFx generates a strongly stable semigroup, where FX = —BR~l (B*X + NC), then it is the only self-adjoint solution with this property. Proof Let Xi <E £(Z) and X2 € £(Z) be two such solutions of (4.1). Let FXi = -BR~l(B*Xi + NC), i = 1,2, and let TFx (t) be the strongly stable semigroup generated by A + BFi. The Riccati equation (4.1) can be rewritten in the following ways:
for all z € "D(A). Subtracting the two equations above, we obtain
This implies that for all Zi,z2& T>(A) there holds
Thus, (TFX ( t j z i , (Xi — XzjTpx (t)Z2) is equal to some constant, which must be 0 by the strong stability of 2>x (t) and TFx (t). Substituting t = 0 and noting that V(A) is dense in Z, we obtain that Xi = X2.
40
Chapter 4. Riccati Equations for Statically Stabilizable Systems
Next, we prove our main result of this chapter. The sufficiency part is proven in Theorem 16 of Staffans [99] by a different approach. Necessary and sufficient conditions can also be found in the report by Mikkola [73]. Theorem 4.3.2 Consider the strongly stable system S(j4,B,C) and the operators N e L(Y, U), Q = Q* e £(Y), R = R*€ L(U), with R boundedly invertible. There exists a unique, self-adjoint, strongly stabilizing solution to the Riccati equation (4.1) if and only if the associated Popov function II has a Wiener-Hopffactorization. Proof Necessity: Suppose that the Riccati equation has a strongly stabilizing solution X with corresponding feedback operator FX- Then, with a long but straightforward calculation, using the fact that X satisfies the Riccati equation, we can check that the following expression holds: We define N(s) = I - Fx(sl - A)~1B, M(s) = RN(s). To show that N and M form a Wiener-Hopf factorization of II, it is sufficient to show that N(s) € H 00 (£(C/)) and N"1 (s) e HOC (£(U)), because the invertibility of R shows that the same then holds for M and M'1. Because A' is a strongly stabilizing solution, the corresponding closed-loop system is input-output stable. Asaconsequence, FX(S/ — Apx)~lB eHoo(£([/))andso N~1(s) = I + Fx(sI-AFx)-lB€U00(£(U)). To prove that N e H ^ ( £ ( U ) ) , note that B*(slA*Ylz eH 2 (C7) implies that (/ - N~(-s))u = B*(sl - A*)~lF^u is in H 2 (C/) for all u € U. Moreover, N(s) = (M"")"1!! € LCO(£(C/)) and consequently N~(-s) 6 Loo(/:(CO), as well. So, by Lemma 4.2.4, N(s) e H*, (£([/)). Sufficiency: By assumption, S(A,B,(7) is a strongly stable system. Suppose that II has a Wiener-Hopf factorization, H(JLL>) — M~(jw)N(jui). By part 3 of Lemma 4.2.3, 7n is boundedly invertible and by part 2 of the same lemma it follows that ~R. is boundedly invertible. As in the proof of Theorem 2.15 in Weiss [111], one can show that the self-adjoint operator X defined by
is in C(Z) and satisfies the Riccati equation (4.1). By the boundedness of all operators involved, it follows that u € L 2 (0, oo; U) and y € L,2(0, oo; Y), where
Because of the C-factor we cannot conclude exponential stability from (4.14), as was done in Weiss [111, 112], but we can show that Tpx is a strongly stable semigroup by considering the perturbation formula from Theorem 3.2.1 in Curtain and Zwart [36]:
Now, since Y,(A,B,C) is a strongly stable system and u(t) e L2(0,oo;[7), Lemma 2.1.3 shows that the right-hand side tends to zero as t goes to infinity. Hence, 1>X (t) is a strongly stable semigroup.
Section 4.3. Riccati Equations for Stable Systems
41
Next, we prove the input-output stability of the optimal closed-loop system, which is the system given in part 2 of Definition 4.1.1. That is, we show that FX(S! - Apx)~lB and C(sl — Apx)~lB € HOC. Let us start with the first one. From Lemma 4.2.7 we have that N has a realization (T,$,$jv.Fjv) as a well-posed linear system. It follows from this that N has A as generator of its semigroup and that B is its input operator. Because B is bounded, N actually has a realization as a regular linear system with generating operators (A,B,C^,D^); see Theorem 5.8 in Weiss [111]. If we normalize N such that limA-xx) N(X)u = u (this can be done because N is invertible), then clearly DN = I. To find CAT, note that K-l(W*Q + N)Vz0 = R~1(B*X + NC)TFx(-)z0 and U = W*MWN. Thus, using the defining equation (4.10) for ~3>N, we obtain Using the regularity of Fjv, with D = I, and taking t — 0 in the above equation we see immediately that So, N(s) — I - Fx(sl - A)~[B. Because AT is a Wiener-Hopf factor, it follows that N-l(s)=I + Fx(sI-AFx)~lBelllx>(£(U)). To prove that also C(sl - AFx}-lB e HOC, we use the perturbation formula C(sl - A)- 1B e HOC (£([/, Y)) because of the strong stability of£(A,B, C) and we have just proved that F\(sl — AF-X)^1B € HOO (£([/)). It now follows from the perturbation formula above that C(sl - Afx)~1B € H 00 (£(t/ i y)). Thus, we have proven the inputoutput stability of the closed-loop system. Next, we prove that the closed-loop system is input stable. To do so, we use the perturbation formula Now, the input stability of E(A,B,C) combined with the input-output stability of the closed-loop system shows that for all z e Z, B*(si- A"FX)~Iz € H 2 (C7). The output stability of the closed-loop system follows directly from (4.13) and (4.14). The uniqueness of the strongly stabilizing solution is proven in Lemma 4.3.1. In the coercive case, we obtain the following corollary. Corollary 4.3.3 Consider the strongly stable system T1(A,B,C} and the operators N, Q, R. There exists a unique, self-adjoint, strongly stabilizing solution to the Riccati equation (4.1) if the associated Popov function II is coercive. Conversely, if there exists a unique, self-adjoint, strongly stabilizing solution to the Riccati equation (4.1) and ifR>fj,I for some fj, > 0 (i.e., R is coercive), then the associated Popov function H is coercive. Proof Assume that II is coercive. Theorem 3.7 in Rosenblum and Rovnyak [90] shows that II has a Wiener-Hopf factorization. Theorem 4.3.2 then implies that (4.1) has a unique, self-adjoint, strongly stabilizing solution. For the converse, assume that R is coercive and that (4.1) has a self-adjoint, strongly stabilizing solution X, with corresponding feedback FX- In the proof of Theorem 4.3.2 it is shown that where S(s) = I - Fx(sl - A)~1B and S^'1 e H CO (£(C/)). Because R is coercive and 5 is invertible over H 00 (£(f/)), II is also coercive.
42
Chapter 4. Riccati Equations for Statically Stabilizable Systems
4.4 Riccati Equations for Unstable Systems Next, we turn our attention to the strongly stabilizing solution for systems that are not stable. The following lemma shows that in certain cases, we can reduce the unstable case to the stable case. Lemma 4.4.1 1. X = X" 6 £-(Z) satisfies the Riccati equation (4.1) if and only if X satisfies the feedback Riccati equation
for all z e T>(A), where AF = A + BF. 2. The Popov function 11 associated with (4.1) is related to the Popov function Up associated with (4.15) by
Proof Part 1 was established as Lemma 23 in Weiss [112] and part 2 as Lemma 24 in the same paper. His proofs are a matter of algebraic manipulations. The main point is that the feedback operator F\ corresponding to (4.1) is related to the feedback operator FXf. corresponding to (4.15) by F\ = F + FXF • To retain the original form of the Riccati equation, we must use feedbacks of the form F = KC. In this case, we can extend the result of the previous lemma, as follows. Lemma 4.4.2 Let F = KC for some K € £(Y, U). X is a strongly stabilizing solution for (4.1) if and only if it is a strongly stabilizing solution for (4.15). Furthermore, the generator of the strongly stable semigroup is the same in both cases. Proof Let us denote Ap = A + BF and let the feedback operator corresponding to (4.1) be FX and the feedback operator corresponding to (4.15) be FXF- They are related by FX = F + FXF • This implies that the generators of the closed-loop semigroups are equal: AF + BFXF =A + BFX• Hence, we need to show that Z(A + BFX, B, [ C" Fx ] *) is a strongly stable system if and only if the system E(A + BFX, B, [ C* Fx - F* ] *) is strongly stable. This follows directly, because we took F — KC. The above lemmas allow us to reduce the problem of a Riccati equation for an unstable system to one for a stable system, provided that the unstable system is statically Stabilizable. The sufficiency part of the following theorem is proven in Theorem 16 of Staffans [99], using a different approach. Theorem 4.4.3 Consider the Riccati equation (4.1), and assume that E(A,B,C) is statically Stabilizable by a static output feedback u = Ky. Let II^c be the transformed Popov function, as in (4.17), with F = KC. 1. Equation (4.1) has a unique, self-adjoint, strongly stabilizing solution if and only if HKC has a Wiener-Hopffactorization.
Section 4.4. Riccati Equations for Unstable Systems
43
2. For the coercive case, define the quadratic form
If there exists an £ > 0 such that for all (w, z, u, y) € R x V(A) x U x Y satisfying juz = Az + Bu, y = Cz, there holds f(y,u) > £(j|y|| 2 + \\u\\2), then the Riccati equation (4.1) has a unique, self-adjoint, strongly stabilizing solution. Conversely, if R > nl for some /j. > 0 and the Riccati equation (4.1) has a unique, strongly stabilizing solution, then there exists an e > 0 such that F(y,u) > £(||y||2 + ||u||2) for all (u, z, u, y) 6 R x V(A) x U x Y satisfying juz = Az + Bu, y — Cz. Proof 1. Theorem 4.3.2 shows that the feedback Riccati equation (4.15) has a unique, self-adjoint, strongly stabilizing solution if and only if UKC has a Wiener-Hopf factorization. Lemma 4.3.1, Lemma 4.4.1 and Lemma 4.4.2 show that the feedback Riccati equation (4.15) has a unique, self-adjoint, strongly stabilizing solution if and only if the Riccati equation (4.1) has one. 2. Let AKC = A + BKC. We introduce a second quadratic form
where QK = Q + K'N + N*K + K'RK and NK = N + RK. The proof will consist of the following three steps: (a) First we show that there exists an £ > 0 such that .F(y,u) > £(||y||2 4- ||u||2) for all (u,z,u,y) e R x V(A) x U x Y satisfying juz = Az + Bu, y = Cz if and only if there exists a 5 > 0 such that J^K(y,v) > <5(||y||2 + \\v\\2) for all (w,z,v,y) e R x T>(A) x U x Y satisfying juz = AKCz + Bv, y = Cz. (b) Second, we show that there exists a 6 > 0 such that Fi<(y,v) > $(\\y\\2 + |H|2) for all (w,z,v,y) € R x T>(A) x U x Y satisfying jujz = AKcz + Bv, y — Cz if and only if HKC is coercive. (c) Finally, we show that there exists a strongly stabilizing solution if UKC is coercive, and conversely that UKC is coercive if there exists a strongly stabilizing solution and R is coercive. (a) Note that if we take u = Ky + v, then jwz = Az + Bu, y = Cz if and only if jwz = AKCZ + Bv, y — Cz. Furthermore, in this case, .F(j/,u) = TK(y, v) and
So, taking a = max(2,1 + 2||K\\ 2 ), we have
Assume now that ^"(y,u) > £(\\y\\2 + ||u 2 ); then
44
Chapter 4. Riccati Equations for Statically Stabilizable Systems
(b) Let UKC be coercive, i.e., there exists £ > 0 such that for all v € U and almost all w e R, (UKC(JI^)V,V) > £\\v\\2. Let (y,v) be such that there exist u> and z satisfying jijz = AKCZ + Bv, y = Cz. It is shown by Huang [54] that since AKC generates a strongly stable semigroup, on the imaginary axis it can have only continuous spectrum. Therefore, for all u> 6 R, (jwl — Axc)^1 exists, though it may be unbounded. Consequently, y can be expressed as y = C(jul — A]
Therefore,
Hence, choosing 8 = £/(c2 +1),
Conversely, if we assume that FK(y,v) > <5(||y||2 + \\v\\2), then for almost all u e M, we have
i.e., UKC is coercive. (c) This part of the proof is a direct consequence of Corollary 4.3.3, Lemma 4.4.1 and Lemma 4.4.2.
4.5
Special Types of Riccati Equations
In this section we show how the general results of Sections 4.3 and 4.4 apply to certain special types of Riccati equations. First we use the results of Section 4.3 to derive a Riccati criterion for J-spectral factorizations and to find versions of the positive real and bounded real lemmas. After that we specialize the result of Section 4.4 to the case of the standard LQ Riccati equation for statically Stabilizable systems.
4.5.1
J-Spectral Factorizations
One important special case of Theorem 4.3.2 is the application to J-spectral factorizations for systems that are strongly stable. In this case, the coercivity assumption (4.4) is not satisfied. Define for Hilbert spaces Z+, Z_ and a real 7 > 0,
Definition 4.5.1 Suppose that G e L00(£([7, Y)), where U = U+ x [7_, Y = Y+ x Uand U+, V+, U- are separable Hilbert spaces. A J-spectral factorization ofG is defined by
where W and W~l € HOQ(£({/)) and W and W~l are regular.
Section 4.5. Special Types of Riccati Equations
45
Theorem 4.5.2 Consider the strongly stable system ^(A,B,C,D) and let U = U+x U- be its input space, Y = Y+x U- be its output space and G(s) be its transfer function. G has a J-spectral factorization if and only if 1. there exists a boundedly invertible operator W^ € £(Y+ x [/_) such that and 2. the Riccati equation (4.1) with the operators R = D*Jy+tjj_ (j)D, Q = JY+,U- (l) and N = D*Jy+iu_ (7) has a unique, self-adjoint, strongly stabilizing solution X. In this case, a J-spectral factor for G is given by
where L = Ju+iU_(7)-*W-*(B*X + D*JY+,u_(f)C). Proof For convenience of notation we write J[j = Ju+,u_ (7) and Jy — JY+,U_ (7)Sufficiency: The formula for W (s) follows by direct substitution. It remains to prove that the spectral factor W and its inverse are in H 00 (£(C/)). It is easier to consider a normalized factor W instead. We define W(s) = W^lW(s) and show that W and its inverse are in H 00 (£(t/)). The Riccati feedback Fx is given by Fx = -(W^0JUW00)-1(B*X + D*JyC) and so we can express W and its inverse in terms of FX as
Because X is a strongly stabilizing solution, the above expression implies that W 1 6 Hoo(£(C/)). By the input-output stability of the system E(A,B,C,D), we have that G~JYG G Loo(£((7)) and consequently that
The input stability of T1(A,B,C, D) allows one to conclude that
From Lemma 4.2.4 it now follows that W(s) and hence W(s) € 'B.00(£(U)). Necessity: Conversely, suppose that G has a J-spectral factorization. We first prove part 1. Let the limit of the spectral factor W along the positive real axis be
Note that in the definition of a J-spectral factorization, we demanded that W be regular. Hence, this limit exists. Then, taking limits on both sides in (4.18), we obtain D*JyD = W^JuW^. For the proof of part 2, we define II — G~JyG. Note that II is the Popov function associated with the Riccati equation (4.1) with Q = Jy, N = D*Jy, R — D*JyD. Setting M(s) = W(s), N(s) = JuW(s), we obtain a Wiener-Hopf factorization of II. Consequently, the Riccati equation has a unique, self-adjoint, strongly stabilizing solution.
46
4.5.2
Chapter 4. Riccati Equations for Statically Stabiliiable Systems
The Strict Positive Real Lemma
Let G(s) = D + C(sl — A)"1B be the transfer function of the strongly stable system E(A,B,C,D). The strict positive real lemma states the following: G is strictly positive real (see Definition 2.2.3) if and only if D + D* >el for some e > 0 and there exists a unique, self-adjoint, nonnegativc, strongly stabilizing solution X to the Riccati equation
for z € T)(A). In this case, there exists a spectral factorization for G + G~ (i.e., there exists M(s) e H 00 (£(f/)) such that M~l(s) e H00(£(C/)) and G + G~ = M~M almost everywhere on the imaginary axis). M(s) is given by
Note that taking the limit along the positive real axis for s -» oo in the inequality G(s) +G(s}* > el shows that the direct feedthrough Z? of a strictly positive real transfer function G(s) = D + C(sl-A}~1B always satisfies D + D* > el. The strict positive real lemma is a direct consequence of Corollary 4.3.3, because G(JUJ) -f G(juj)* is the Popov function corresponding to
and X is a self-adjoint stabilizing solution of (4.20) if and only if -X is a self-adjoint stabilizing solution of (4.21). We still need to prove that X > 0. Note that the Riccati equation (4.20) corresponds to N = -/, Q = 0 and R = -D - D*. Using (4.12), we obtain
Now, since G is strictly positive real, we have F + F* > el, where F is the extended inputoutput map of E(A,B,C, D). So, Tl = -F - F* < -el and X > 0. Note that under the extra assumption that the system is approximately observable (i.e., ker(^) = {0}) we obtain that X > 0.
4.5.3
The Strict Bounded Real Lemma
A transfer function G(s) is bounded real if
When the inequality is strengthened to / — G*(s)G(s) > el for some e > 0, then G(s) is strictly bounded real. Note that if E (A, B, C, D) is strictly bounded real, then automatically also I — D*D>sI. This follows by taking the limit as s -> oo along the positive real axis in the inequality J - G*(s)G(s) > el. Let S(A,B,C,D) be a strongly stable system with transfer function G(s). The strict bounded real lemma then reads: G is strictly bounded real if and only ifI — D*D>eI for some e > 0 and there exists a unique, self-adjoint, strongly stabilizing solution X to
Section 4.5. Special Types ofRiccati Equations
47
for z e T>(A). M(ju})*M(juj),
=
In this case there exists a spectral factorization / - G(j<jj)*G(j(j) which is given by
Furthermore, in this case X > 0 and if £(.A, — ,C) is approximately observable, then X even satisfies X > 0. The strict bounded real lemma is a direct consequence of Corollary 4.3.3, because 7 - G(ju)*G(juj) is the Popov function associated with
and X is a stabilizing solution of (4.22) if and only if -X is a stabilizing solution of (4.23), in a similar way as for the strict positive real lemma. That X > 0 follows from the construction in (4.12): X = <£*(/ + F(7 - D*D)~1W*)'$, where F is the extended input-output map of S(v4,B,C,D). Clearly,X>0bythepositivityof7-D*D. NotethatX >OifS( J 4,-,C) is approximately observable.
4.5.4
The Coercive LQ Case
Let H(A,B,C) be a statically stabilizable system and let
for some a > 0. This coercivity assumption is satisfied, for instance, in the standard LQ case, where N = 0, Q = /, R = I, or in the case that N = 0, R > 51 and Q > /j.1 for some (i > 0,6 > 0. Under these assumptions, clearly we have f(y,u) > a(||y||2 + ||u||2). Hence, by part 2 of Theorem 4.4.3, the corresponding Riccati equation will have a unique, self-adjoint, strongly stabilizing solution X. As an application of this result, we look at the standard LQ control problem for a statically stabilizable system with direct feedthrough. Theorem 4.5.3 Consider the standard LQ control problemfor the statically stabilizable system H(A,B,C,D). That is, consider the problem of minimizing
subject to
The Riccati equation associated with this problem is given by
for z 6 TJ(A), and it has a unique, self-adjoint, strongly stabilizing solution. Proof For this LQ problem, Q = /,N = .D*,.R = / + £>".Dandit follows immediately that (4.26) is the corresponding Riccati equation.
48
Chapter 4. Riccati Equations for Statically Stabilizable Systems Note that, even though Q and R are coercive, in general
is not coercive. However, we can transform the problem into an equivalent one without cross-terms in the cost and for a system without direct feedthrough. To do so, we define y = (I + DD*)~?Cz and apply the state feedback
Thus, we obtain the following equivalent LQ control problem:
subject to
where A = A - B(I + D*D)~1D*C, B = B(I + D*D)-? and C = (I + DD*)~?C. It can easily be checked that %(A,B,C) inherits the static stabilizability property from E(A,B,C). In this new LQ problem, Q = I, R = I and N = 0, so condition (4.24) is satisfied and the Riccati equation
has a unique, self-adjoint, nonnegative, strongly stabilizing solution. From the fact that (I + DD*)~l = I + D(I + D*D)'1 D* it is easy to see that the two Riccati equations (4.26) and (4.27) are exactly the same. Hence, (4.26) also has a unique, self-adjoint, nonnegative, strongly stabilizing solution. D Dually, it can be shown that if 'S(Ax,C*,Bf,D*) is statically Stabilizable, then the standard filter Riccati equation for z e T>(A*),
has a unique, self-adjoint, nonnegative, strongly stabilizing solution. Similar to what was presented in Section 3.3, this means that the associated closed-loop system E(^4* + C*Ly,C*, [ B LY ]*,£)*) is a strongly stable system. For filtering, one would be interested in the stability of the system S(>1 + Ly C1, [ B Ly ] ,C,D) and we do obtain input stability, output stability and input-output stability for this system. However, in general A + LyC will not generate a strongly stable semigroup, although its adjoint does. We should remark that the result in this section is weaker than the result in Theorem 3.3.1. In the same way as here, we could include a direct feedthrough term into that result. Hence, the only difference between the two results is that we need to impose static stabilizability here, which is stronger than the combination of strong stabilizability and strong detectability that was assumed in Theorem 3.3.2.
Section 4.6. Examples
4.6
49
Examples
In this section, we will illustrate the results of the preceding sections by applying them to a number of problems. Rather than considering the most general class of systems possible, we specialize to systems of the T.(A,B,B*) class, which was introduced in Section 2.2. It will turn out that the conditions for the existence of solutions of the Riccati equations can in many cases be easily checked for this class of systems. We will consider Riccati equations for systems Y,(A,B,B*,D) under the assumptions A1-A5 (see page 17).
4.6.1
The Strict Positive Real Lemma
We consider the strict positive real lemma for systems E(AB,B,B*,D), where Ag — A — BB* and D + D* > el for some e > 0 under the assumptions A1-A5. In this case, the transfer function G(s) = D + B*(sI — AB)~IB is strictly positive real. Therefore, by the result of Section 4.5.2, the Riccati equation for z £ T>(A),
has a unique, self-adjoint, strongly stabilizing solution X. Furthermore, there exists a spectral factorization
where the spectral factor -M is given by
4.6.2
The Standard LQ Riccati Equation
For the system Y^(A,B,B*,D) satisfying assumptions A1-A5, we consider the standard LQ optimal control problem, i.e., we want to minimize the criterion (4.25). From the example in Section 4.5.4, it follows that the associated Riccati equation for z £ ~D(A),
has a unique, strongly stabilizing solution X. This case was also studied in Curtain and Zwart [37] for the same class of systems.
This page intentionally left blank
Chapter 5
Numerical Approximation Controllers designed on the basis of an infinite-dimensional model of the plant are very often infinite-dimensional, as well. In particular, this is the case for LQ-, LQG- and Hoccontrollers. Of course, when one wants to actually compute or even implement such a controller, one has to approximate it by a finite-dimensional controller. Therefore, over the years much research has been directed toward developing approximation schemes for these controllers and, in particular, approximation schemes for numerical solutions of algebraic Riccati equations. Many of the papers use an approach based on approximation results for Co-semigroups, i.e., they use a version of the Trotter-Kato theorem in the proof of convergence of the solution of the Riccati equation (see, for instance, Banks and Burns [11], Gibson [49], Ito [55], Kappel and Salamon [58] and the references therein). In all of these papers it was assumed that A generates an exponentially stable semigroup or that the pair (A,B) is exponentially stabilizable. Therefore, after the results on the existence of the strongly stabilizing solution of Riccati equations in Chapters 3 and 4, it is logical to next consider the approximation of those strongly stabilizing solutions by finite-dimensional ones. In this chapter we restrict ourselves to the approximation of the strongly stabilizing solution of the standard LQ Riccati equation.
5.1 The LQ Riccati Equation for Strongly Stabilizable Systems As in Chapter 3, we consider the following optimal control problem: For the system
we want to find an input u e LZ (0, oo; U) that minimizes the quadratic cost criterion
In Theorem 3.3.2, we proved that strong stabilizability of %(A, B, C) is sufficient for the existence of a minimizing control u and a self-adjoint, nonnegative solution to the algebraic Riccati equation
57
52
Chapters. Numerical Approximation
for z € 'D(A). In the same theorem, it was proven that if, in addition, ^(A,B,C) is strongly detectable, then the solution X is a strongly stabilizing solution of (5.3) and is the only self-adjoint solution with this property. Next we show how to reformulate the minimization problem for a strongly stabilizable system as an equivalent one for a strongly stable system. We do this in a different way from that in Chapter 4 and, as a result, we obtain a different formula for the strongly stabilizing solution X. The conditions that we impose this time on stabilizability and detectability of the system are stronger than those in Chapter 3, but weaker than those in Chapter 4. We elaborate on the differences at the end of this section. Theorem 5.1.1 Consider the problem of minimizing the cost functional (5.2) subject to Y,(A,B,C) under the assumption that there exist F 6 £(Z,U) and L e C(Y,Z) such that Cl. A + BF and A + LC generate strongly stable Co-semigroups 7>(i) and Ti(t), respectively; is input-output stable and output stable; is input stable, output stable and input-output stable. Then this problem is equivalent to that of minimizing the cost functional
subject to
The optimal input for the optimal control problem (5.4), (5.5) is given by
where WF and ¥p are the extended output map and the extended input-output map, respectively, of the system
Moreover, there exists a unique, self-adjoint, nonnegative, strongly stabilizing solution to the algebraic Riccati equation (5.3), which is given by
Proof (a) Note that with u = v + Fz, the systems (5.1) and (5.5) have the same solution
and JF(V,Z(J) = ^(U^Q). Therefore, the two optimization problems are equivalent if the following two statements are equivalent:
Section 5.1. The LQ Riccati Equation for Strongly Stabilizable Systems
53
It follows from assumption C2 that ?; € L2(0,oo;{7) if and only if J~F(V,ZQ) < oo. Hence, it suffices to prove that v € L2 if and only if u e L2 and J"(M,ZO) < oo. Suppose first that v e L 2 (0,oo;f7). Since E(A + BF,B,F) is input-output stable, Fz(t)-FTp<(t)z0 € L2(0,oo;E7) and since E(A + BF,B,F) is output stable, FTF(t)z0 and hence -Fz(t) e L 2 (0,oo;t/). Thus, u € L 2 (0,oo;J7). The output stability and inputoutput stability of E(A + BF, B,C) imply that Cz(t) € L2(0, oo; Y) which, together with u 6 L2(0,oo;C7), show that J(u,z0) < oo. Conversely, suppose that u G L2(0,oo; U) and J(u,z0) < oo and write
Then Fz(t) - FTL(t)z0 e L 2 (0,oo;U), since I,(A + LC,[B L ] ,F) is input-output stable. In addition, FTL(t)zQ e L2(0,oo;£7), since Z(A + LC,B,F) is output stable. SoFz(t) e L 2 (0,oo;f7) and therefore also v = u-Fz e L 2 (0,oo;C7). (b) We now obtain an explicit solution of the minimization of (5.4) for the stable system (5.5) in terms of the bounded maps ¥p, ^F given by
Since JF(V,ZQ) < oo for all v e L2(0, oo; U), we can write (5.4) equivalently as Next, we show that the cost functional JF is coercive. Let the operator WLF denote the input-output map of the system E(A + LC, [B -L], -F, [ I 0 ]). By assumption C3, WLF e £(L2(0,oo;{7 x y),L 2 (0,oo;E/)). We show that WLFWF = /i2{o,oo;i/). "sing a frequency domain argument. Let ~ denote Laplace transforms and let v 6 L 2 (0,oo;£7), y&L2(Q,oo;Y). Then,
and
Now,
Consequently, (¥LF¥Fv)(t)
= v(t) for all v e L 2 (0, oo; U).
54
ChapterS. Numerical Approximation We deduce further that
Note that, because of the identity WLF^F — I, we cannot have ||FLF|| = 0. Thus,
We can now apply the theory of paragraphs 1.1.1—1.1.3 in Lions [69] to show that there exists a unique minimizing control v € 1^(0, oo; U), which is characterized by
Because (5.8) implies that W^Wp is boundedly invertible, we can solve (5.9) for v to obtain (5.6). From Theorem 3.3.2 we know that
and (5.7) follows. That X is the unique, self-adjoint, strongly stabilizing solution also follows from Theorem 3.3.2. D It is interesting to note that although the strategy in Section 4.4 is also to express the general control problem via a feedback in terms of an equivalent stable one, the feedbacks are different and the formulas for X are very different. In the stable case (F = 0, L = 0), however, the formulas for X coincide. For proving convergence results for the standard LQ Riccati equation the approach in this chapter is more convenient. It turns out that the conditions Cl, C2, C3 are key to proving our main approximation result in Section 5.3. Consequently, we introduce the following definition. Definition 5.1.2 The system %(A,B, C) is strongly stabilizable-detectable if there exist F £ C(Z, U) and L 6 C(Y, Z) such that Cl. A + BF and A + LC generate strongly stable Co-semigroups TF(t) and TL(t), respectively; is input-output stable and output stable; F) is input-output stable, input stable and output stable. Let us point out the difference between a system that is strongly stabilizable and strongly detectable and a system that is strongly stabilizable-detectable. As a short inspection reveals, every strongly stabilizable-detectable system is both strongly stabilizable and strongly detectable; the only candidates for F and L in Definition 5.1.2 are F's which strongly stabilize the system and L's which strongly detect the system, in the sense of Definition 2.1.7. However, the operators L and F have to be related to each other in some sense: in condition C3 of Definition 5.1.2, both F and L play a role. The output stability and input-output stability requirements will put a restriction on the possible combinations of F and L. Second, the input-output stability in condition C2 is, in general, not satisfied if F only strongly
Section 5.2. Approximation of Strongly Stable Systems
55
stabilizes the system. Note, however, that the concept of strong stabilizability-detectability is a weaker notion than that of static stabilizability. We remark that if both A + BF and A + LC generate exponentially stable Co-semigroups on Z, then the conditions Cl, C2 and C3, above, hold automatically.
5.2 Approximation of Strongly Stable Systems In this section we show how to approximate a strongly stable system by a sequence of finitedimensional systems. We show that our approximation scheme guarantees the convergence of the extended output maps and the extended input-output maps. In Section 5.3, we use these results to prove the strong convergence of a sequence of stabilizing solutions of finitedimensional Riccati equations to the strongly stabilizing solution of the infinite-dimensional standard LQ Riccati equation. Consider again the system on the Hilbert spaces Z, U, Y,
This infinite-dimensional system is approximated by a sequence of systems on the finitedimensional spaces ZN = Rk^, UN = R m < N >, YN = R p W,
We assume the existence of injective linear maps
and surjective linear maps
such that TTNIN, pNjN and aNkN are identity maps and iN-nN, JNpN and kNaN are orthogonal projections. Note that it is not necessary that (iN)* = TTN , (jN)* = pN or (kNY =aN. On the spaces M f e ( j V ) ,R m ( J V ) andR^W we will always consider the induced inner products
(AN)*, (BN)*, (CN)* will denote the adjoint matrices with respect to the induced inner products. Because we always use the induced inner products on Rk(N\ Rm W and Rp(N1,
56
ChapterS. Numerical Approximation
it is obvious that ||»" || = IIJ^H = Hfc^ll = UTT^H = \\pN\\ = \\aN\\ = 1. Define ZN = i&nge(iNirN), UN = range(jNpN) and YN = iange(kNaN). Then,
and similar equalities hold for all operators defined in an analogous way. Let r(-),,\f,F,G r (s) be the semigroup, extended input map, extended output map, input-output map and transfer function, respectively, associated with Y,(A,B,C), and let TN(-),^N,^N,WN,GN(s) be the same operators but associated with the system E(AN, BN', CN). In the latter case, we are dealing with finite-dimensional systems. Hence, TN(t) = eA * and (*f?Nz)(i) = CNeA *z. Let us define the notion of strong convergence of a sequence of systems. Definition 5.2.1 A sequence of systems H(AN ,BN,CN) converges strongly to the system T,(A,B,C) if, for all ztZ,
uniformly on compact time intervals and, for all z € Z, u € U, y € Y,
Note that the convergence of the semigroup in the above definition implies, by taking t - 0, that for all z e Z,
In connection with the convergence of a sequence of finite-dimensional systems to an infinite-dimensional system, we introduce a notion of uniform stability of the sequence of finite-dimensional systems. Definition 5.2.2 A sequence of systems %(AN, BN, CN) is uniformly output stable (/ there exists an M > 0 such that for all z € Z,
Section 5.2. Approximation of Strongly Stable Systems
57
A sequence of systems Y,(AN ,BN ,CN) is uniformly input-output stable if there exists an M > 0 such that
In Lemmas 5.2.4 and 5.2.6, we show the conditions under which the extended output maps and extended input-output maps of a sequence of finite-dimensional systems converge to their infinite-dimensional counterparts. In the proof of these lemmas, as well as in all other convergence proofs in this chapter, we will use the fact that if the sequences of operators RN and SN converge strongly to R and S, respectively, then RNSN converges strongly to RS. Often, in our convergence results we will have to prove the convergence of a sequence of operators, operating on £2(0, oo). In these cases, the following lemma allows us to restrict our attention to the dense subset of 1/2 functions with compact support. Lemma 5.2.3 Let X\ and X2 be Banach spaces and let R^ 6 £ ( X i , X2) for all n e N. Let S be a dense subset of X\ and let R € £.(X\, X%). Assume that 1. there exists an M > 0 such that for all n, \\Rn\\ < M, and
The proof of this lemma is straightforward, and hence it is omitted. Lemma 5.2.4 Suppose that T(t) is strongly stable, Y,(A,B,C) is output stable, the sequence of finite-dimensional systems H(AN ,BN ,CN) converges strongly to T,(A,B,C) and E(AN ,BN,CN) is uniformly output stable. Then, for all ztZ,
Now we derive estimates for each of the integral terms 01,02,03,04. For ai, we have
55
Chapters. Numerical Approximation
The right-hand side tends to zero as N tends to infinity by the Lebesgue dominated convergence theorem because the systems E(AN, BN, CN) are strongly convergent to E(A, B,C). 0*2 satisfies
which tends to zero as t\ tends to infinity, independently of N, because of the output stability of I,(A,B,C) and the strong stability ofT(t). Since the systems Z,(AN,BN,CN) are uniformly output stable, there exists a positive constant M such that
The right-hand side tends to zero as N tends to infinity because 'E(AN ,BN,CN) converges strongly to E(vl, B, C). For the same constant M, we have
the right-hand side of which tends to zero, independently of N, because T(t) is strongly stable. These estimates together show that kN^N/nNz -»\tz strongly as N -> oo. Because \\iN(yN)*aN\\ = || \&N || <M, it follows from Lemma 5.2.3 that it is sufficient for the proof of convergence of iN(^N)*aNy to $!*y to consider y e L 2 (0,oo;y) with compact support. So, lety 6L 2 (0,oo;F) have support on [0,ii], t\ > 0. Then,
Both integrals tend to zero as TV —> oo because the systems ~S(AN,BN,CN) are strongly convergent to £(A, B, C). Corollary 5.2.5 Under the assumptions of Lemma 5.2.4,
forallzeZ. Proof This result follows directly by taking Laplace transforms in Lemma 5.2.4. Lemma 5.2.6 Suppose that the assumptions of Lemma 5.2.4 hold and, in addition, ^(A,B,C) is input-output stable and E(AN, BN,CN) is uniformly input-output stable. Then, for all u € L2 (0, oo; U),
in L2(0, oo; Y). Similarly, for all y € L2(0, oo; Y),
i«L 2 (0,oo;C/). Proof By the uniform input-output stability of the systems E(AN,BN,CN), the operators kNWNpN are uniformly bounded in N. Therefore, by Lemma 5.2.3 it is sufficient
Section 5.2. Approximation of Strongly Stable Systems
59
to prove the convergence WNu —» Fu for input functions u with compact support. Let u € L 2 (0,oo;C7) with support in [0,^], let y,yN be denned by y = Fu, yN = kNWNpNu and define MI = sup^sup t6 r 0it] ] Hi^e" 4 'TT^H. Then,
for 0 < t < ti. By the strong convergence of E(AN,BN,CN) to E(A,B,C) and the Lebesgue dominated convergence theorem, it now follows in the same way as in Kappel and Salamon [58] that \\y — yN\\L2(o,t1-,Y) —> O a s ^ V — > oo. Furthermore, we define
By the strong convergence of E(AN,BN,CN) to T.(A,B,C), we have that z£ -> zu as N —> oo. In addition, since u has support in [0,£i], we obtain for t>t\,
Lemma 5.2.4 is now used to prove that ||y — yN\\L2(t\,oo-Y) —> 0 as JV —> oo. To prove the convergence of (F^)*, note that because of the uniform boundedness of (¥N)* it is again sufficient to only consider y with compact support [0,
forallu(s)£H2(U).
Similarly, for all y(s) € H2(Y),
Proof This result follows directly by taking Laplace transforms in Lemma 5.2.6. We remark that the above properties even hold for u(s) = u, y ( s ) = y (see Corollary 5.2.5). However, in general we do not have convergence in HOO, i.e., we do not have
see Proposition 1 and the counterexample on pages 1146-1147 in Kappel and Salamon [58].
60
ChapterS. Numerical Approximation
5.3 Approximation of the Strongly Stabilizing Solution In this section, we prove our main convergence result for Riccati equations for strongly stabilizable-detectable systems. The idea will be to transform the LQ problem to an equivalent stable problem as in Theorem 5.1.1. Then, the resulting stable system is approximated, as was done in the previous section. The convergence results of Section 5.2 then will guarantee the convergence of the Riccati solutions. Without proof, we remark that, as in Kappel and Salamon [58], it can be shown that if a sequence of systems Y.(AN,BN,CN) converges strongly to Z(A,B,C) and matrices FN £ R">Wx*W are chosen such that the operator sequences jNFNnN £ £(Z,U) and iN(FN)*pN € £(Z, U) converge strongly to F and F*, respectively, then strong convergence of the system holds, as well. Theorem 5.3.1 Consider the algebraic Riccati equation (5.3), and assume that 1. the system is strongly stabilizable-detectable (i. e., there exist. and L £ £(Y, Z) for which the properties C1-C3 in Definition 5.1.2 are satisfied); 2. there exists a sequence of systems 3. there exists a sequence of matrices a.
strongly and erator from part 1;
b.
that is strongly convergent to such that strongly, where F is the op-
is a stable matrix;
c.
is uniformly output stable;
d.
is uniformly input-output stable;
4. there exists a sequence of matrices a. b.
such that
is a stable matrix; is uniformly input-output stable.
Then, for every
where X £ £(Z) is the unique, self-adjoint, nonnegative solution of the algebraic Riccati equation (5.3) and XN € £(M.k^) are the unique, self-adjoint, nonnegative solutions of the sequence of algebraic Riccati equations
Moreover, denoting loop systems
the approximating closed-
Section 5.3. Approximation of the Strongly Stabilizing Solution
61
converge strongly to the infinite-dimensional closed-loop system
Proof We apply Theorem 5.1.1 to the approximating sequence of finite-dimensional systems ^(AN,BN ,CN). Since AN +BNFN and AN+LNCN are stable matrices, all the assumptions C1-C3 are satisfied and the solutions to (5.13) are given by
where \?^ and ¥p are the output map and input-output map, respectively, of the system
We apply Lemma 5.2.4 to obtain
for all z € Z and w € L2(0,oo;t/ x Y), where jN (&kN denotes the direct sum of jN and kN defined by (JN ®kN)(uN ,yN) = (jNuN,kNyN). Similarly, Lemma 5.2.6 yields
as N —>• oo. Analogously to the definition of ^LF in the proof of Theorem 5.1.1, we define F^p to be the input-output map of the system
Assumption 4b of the present theorem tells us that ||F^|| is uniformly bounded. Now, from assumption 3d, sup ||FJ^|| < oo and, from (5.8), we have
This shows that ||((F^)*F^)- 1 || is bounded from above uniformly in TV. We also deduce that \\jN'((^^-)*¥N)~lpN\\ is uniformly bounded from above, the reason being that we use the induced inner products on R m ( N ) and so
Thus,
as
62
Chapter5. Numerical Approximation
The representations of X and XN ((5.7) and (5.14), respectively), together with the convergence results in (5.15)-(5.19), show that iNXNirN —> X strongly as N -» oo. It is an easy consequence of the strong convergence of the stabilizing solutions that the optimal feedback operator F$? and its adjoint converge strongly to their infinite-dimensional counterparts. The convergence of the approximating closed-loop systems now follows from the remark just before Theorem 5.3.1. In the above theorem we have established the strong convergence of the approximating closed-loop systems to the infinite-dimensional closed-loop system. In practice, however, one would like to apply the finite-dimensional feedback to the infinite-dimensional plant. It is easy to see that the resulting closed-loop system
does converge strongly to the infinite-dimensional closed-loop system as TV tends to infinity. However, we have not been able to show that the resulting system is strongly stable, nor that A + BjN F£NTTN generates a strongly stable semigroup. It probably does not do so, in general.
5.4 Applications to Dissipative Systems with Collocated Actuators and Sensors We consider linear systems E (A, B, B*) under the assumptions A1-A5 that were introduced in Section 2.2. It will turn out that if we choose the approximations of such a system appropriately, i.e., such that the structure is retained in the approximating systems, then most of the assumptions for convergence are automatically satisfied. The algebraic Riccati equation associated with this system is
for all z € 'D(A). It follows from Theorem 3.3.2 that there exists a unique, self-adjoint, nonnegative solution X to the algebraic Riccati equation (5.20). We show here that \\X\\ < 1. Indeed,
which implies that \\X\\ < 1. In the remainder of this chapter, we assume U and Y to be finite-dimensional, U — Y = R m . Furthermore, we assume the existence of a sequence of injective linear maps iN •. R n W ->• Z and a sequence of surjective linear maps TTN : Z -» R n W such that TTNIN = In(Nj and iN7f^ is an orthogonal projection on Z. We define the following sequence of approximating systems on R n W;
Lemma 5.4.1 Let E(AN,BN,CN) be defined as above. Then, it satisfies AN + (A } < 0 and CN = (BN)*. Moreover, E(AN - BN(BN)* ,BN ,CN) is uniformly N f
Section 5.4. Dissipative Systems with Collocated Actuators and Sensors
63
output stable and uniformly input-output stable. If H(AN ,BN ,(BN)*) is observable, then AN - BN(BN)* is a stable matrix. Proof Wefirstshow that CN = (BN)* and AN + (AN)* < 0. For u € K m , z € Kn W,
is an orthogonal projection
and so
Next, consider for
is an orthogonal projection
is an orthogonal projection
and so
Now,
since
is an orthogonal projection
since A is dissipative. Together with this shows that for every N, satisfies properties P4 and P6 of Lemma 2.2.6; hence it is uniformly output stable and uniformly input-output stable. The stability of A follows just as in the infinite-dimensional case from the observability of the pair and . Consider solutions of Thus,
for arbitrary z0 e Rn W. We introduce as a Lyapunov function for this differential equation V(z) — \\z\\2. Then, if we differentiate V along solutions of the differential equation, we obtain
64
Chapter 5. Numerical Approximation
where the inequality follows from the dissipativity of AN. Now, by Lasalle's invariance principle (see Lasalle [61]), all solutions z(t) converge to the largest AN — BN (BN)*invariant subset of the set
Now, let 0 ^ z0 6 S; then, for all t > 0, z ( t ) = e(AN~BN(BN)')tZo
£ s And so> by (5 22)(
hence, for all t > 0, (BN)*e(-A ~B (B )*^z0 = 0 which, by the observability assumption, implies that z0 = 0. Thus, S = {0} and consequently AN - BN'(BN)* is stable. Finally, we give sufficient conditions under which the sequence of finite-dimensional systems J}(AN,BN,(BN)*), as defined earlier in this section, converges strongly to the infinite-dimensional system £(j4,5,.B*). Lemma 5.4.2 Assume that the following conditions are satisfied. 1. The sequence ofinjective maps iN : R n W —>• Z and the sequence ofsurjective maps TTN : Z —> R™^) satisfy irNiN = In(N), iNKN is an orthogonal projection on Z and iNKNz -» z as N -> oo for all z€Z. 2. For all z € T>(A), there exists a sequence ZN € M"^' such that
3. For all z € V(A*), there exists a sequence ZN £ Rn(-N)
sucn
that
Then T,(AN,BN,CN) defined by (5.21) converges strongly to Z(A,B,B*). Proof From Theorem 3.1.7 of Davies [38, p. 80], we have that (5.23) is equivalent to
uniformly on compact time intervals for all z € Z, and (5.24) is equivalent to
uniformly on compact time intervals for all z 6 Z, Next, iN BN — IN-KNB —» B as TV ->• oo and CNirN = (BN)*KN = B*iNirN -> B* as N -> oo, which completes the proof.
Section 5.4. Dissipative Systems with Collocated Actuators and Sensors
65
The following theorem, which is the main result of this section, is now a direct consequence of Theorem 5.3.1, Lemma 5.4.1, Lemma 5.4.2 and the properties of systems X(A,B,B*). Theorem 5.4.3 Let the system ^(A, B, B*) satisfy assumptions A1-A5 of Section 2.2 and, furthermore, assume that U = Y = Rm. Define the sequence of approximating systems Z(AN,BN,CN) as in (5.21). Assume that for all N, Z(AN,BN,CN) is observable and, furthermore, that the assumptions 1-3 from Lemma 5.4.2 hold. Then XN —> X strongly as N —> oo, where X and XN are the solutions of the Riccati equations associated with X(A,B,B*) andZ(AN,BN,CN), respectively. Proof By the assumptions 1-3 of Lemma 5.4.2, we have that the sequence of systems E(AN,BN,CN) as defined in (5.21) converges strongly to E(A,B,B*). By Lemma5.4.1, the sequence of systems 32 (AN — BN (BN)* ,BN ,CN] is uniformly output stable and uniformly input-output stable and, in addition, by the observability assumption these systems are stable. As I2(A,B.B*) is statically stabilizable with K = —I, it is obviously strongly stabilizable-detectable and we can take F = -B* and L = -B. Taking FN = -(BN)* and LN — —BN, obviously all assumptions of Theorem 5.3.1 are satisfied. Therefore, we can conclude that the stabilizing solutions XN of the finite-dimensional Riccati equations associated with Y,(AN,BN,CN) converge strongly to the strongly stabilizing solution X of the Riccati equation associated with 32(A, B,B*). In Chapter 9, we describe a model of propagation of sound in a one-dimensional waveguide, terminated by linear oscillators. There we show that it has a state-space realization in our Y,(A,B,B*) class. This allows us to apply the theory of this chapter to that physical example. Therefore, we describe the numerical implementation of an approximation scheme for this example and we present numerical results concerning the approximation of the strongly stabilizing solution of the Riccati equation. The class of dissipative systems with collocated actuators and sensors has a very special structure. However, one can also allow hybrid systems composed of a distributed system of the structure H,(A,B,B*) coupled with a finite-dimensional system Y.(Af,Bj,Cp). The composite system
will satisfy the assumptions of our theory, provided that the approximating sequence of systems H(AN,BN,CN) is chosen as before and the finite-dimensional subsystem T,(Af,Bf,Cf) is minimal. This broadens the class of possible applications. An example of a system with this hybrid structure is the model for a flexible structure with rigid body dynamics in Section 9.4.
This page intentionally left blank
Chapter 6
Coprime Factorizations and Dynamic Stabilization In this chapter, we develop a theory of strong stabilization by dynamic compensators. Because dynamic stabilization is strongly related to the existence of coprime factorizations, we first treat coprime factorizations for strongly stabilizable systems in Section 6.2. In particular, we are interested in finding state-space formulas for coprime factorizations. For both problems, we have not obtained a satisfactory generalization of the finite-dimensional results, and it is not clear that this is possible.
6.1 Problem Formulation This chapter deals with two important system theoretic problems for strongly stabilizable systems. The first problem is that of finding state-space formulas for coprime factorizations of a given strongly stabilizable system E(A,B,C,D). Our definition of a coprime factorization is taken from Curtain, Weiss and Weiss [34]. They developed a theory of coprime factorizations for the very general class of regular linear systems. The generality of this class lies in the fact that it allows for very unbounded input and output operators. On the other hand, they considered only exponentially stabilizable systems. Definition 6.1.1 Let G be a well-posed transfer function (i.e., G is analytic on some right half-plane). A left-coprime factorization ofG over Hx is a factorization where N e H00(£((7,F)), M e H00(£(y))) such that M(s) is invertible on some right half-plane and for which there existY € H00(£(F,[/)) andX e Hoo(£(y)) such thatfor
seC+,
This factorization is a normalized left-coprime factorization if, in addition,
A right-coprime factorization ofG over HOQ is a factorization
67
68
Chapter 6, Coprime Factorizations and Dynamic Stabilization
where N 6 Hoo(£(C/,y)), M e HX(£(U)), such that M(s) is invertible on some right half-plane and for which there exist Y e H^ (£(F, U)) and X 6 HOO (£(U}) such that for
seC+,
The right-coprime factorization is normalized if, in addition, A factorization is a doubly coprime factorization of'G over H^ // (M,N) and (M,N) are a right- and left-coprime factorization ofG, respectively, and there exist X, Y, X,Y £ H^ such that
If, in addition, (6.2) and (6.4) are satisfied, then the factorization is a normalized doubly coprime factorization. The equations (6.1) and (6.3) are called the Bezout identity, and the functions X, Y, X, Y are the Bezout factors. The second problem considered in this chapter is that of the construction of dynamic compensators that stabilize a given, strongly stabilizable system £(A,B,C,D). For this, we consider the feedback interconnection of a plant EG with a compensator E#, as in Figure 6.1. Let EG and EX have transfer function G(s) and K(s), respectively. We have to restrict our consideration to those transfer functions that give rise to a well-posed interconnection. Definition 6.1.2 A well-posed transfer function K(s) is an admissible feedback transfer function forG(s) if LI — G(s)K(s) is invertible on some right half-plane, and 2. (I — G(s)K(s))~1 is a well-posed transfer function. In the case that K is an admissible feedback transfer function for G, then (/ — KG) ~1 = I + K(I - GK)~1G also exists and is a well-posed transfer function. The closed-loop transfer function from co\(vi,v^) to col^,y^) is given by
Figure 6.1: Configuration for dynamic compensation.
Section 6.2. Coprime Factorizations
69
Often, problems of dynamic stabilization are formulated on the level of transfer functions. In this case, the following notion of input-output stabilization is considered. Definition 6.1.3 Let G and K be well-posed transfer functions. K stabilizes G in the input-output sense if 1. K is an admissible feedback transfer function for G, and 2. the closed-loop transfer function from co\(vi,V2) to col(yi,7/2) is an element of HooOCCtfxy.yxE/)). The input-output stability is often (equivalently) defined in terms of the transfer function from col(i>i,i>2) to col(«i,U2). The strong connection between coprime factorizations and dynamic stabilization is well known. First of all, it is known from Smith [95] that a system can be stabilized (in the sense of Definition 6.1.3) if and only if its transfer function has a coprime factorization over HOC. A second instance of the relation is the celebrated Youla parameterization of all stabilizing controllers for a given transfer function (see Maciejowski [71] or any other book on multivariable control). We quote two results from Weiss and Curtain [113] about the relation between coprime factorizations over H^ and dynamic stabilization for well-posed transfer functions. Lemma 6.1.4 Let G and K be well-posed transfer functions taking values in £(U, Y) and£(Y, U], respectively, and suppose that G has a left-coprime factorization G = M~1N and K has a right-coprime factorization K — UV~l. Then K stabilizes G in the inputoutput sense if and only if MV — NU is invertible over HQQ. Theorem 6.1.5 (Youla Parameterization) Suppose that the well-posed transfer function G has a doubly coprime factorization over HQQ as in Definition 6.1.1. 1. Assume that K is a well-posed transfer function that stabilizes G. Then K has a right-coprime factorization over HQQ if and only if it has a left-coprime factorization over HOC. In this case, there exists a unique Q e HQQ such that X + NQ and X + QN have well-posed inverses and are right- and left-coprime factorizations of K. 2. Conversely, ifQ£ H^ is such that one of X + NQ or X + QN has a well-posed inverse, then the other has a well-posed inverse, as well, and the equality in (6.7) holds. In this case, K defined in (6.7) stabilizes G. In Section 6.3, we deal with the problem of finding stabilizing dynamic compensators for strongly stabilizable systems, i.e., we try to find systems with transfer function K that stabilize a given strongly stabilizable system with transfer function G in the input-output sense. Furthermore, we study the problem of whether the resulting closed-loop system is actually a strongly stable system. In Chapter 7, we use the theory of this chapter to solve the problem of robust input-output stabilization with respect to coprime factor perturbations.
6.2 Coprime Factorizations Our aim in this section is to give state-space formulas for doubly coprime factorizations of strongly stabilizable systems. Our treatment of coprime factorizations will stay close to the approach in Curtain, Weiss and Weiss [34]. In that paper, they considered state-space formulas for coprime factorizations of transfer functions of regular linear systems.
70
Chapter 6. Coprime Factorizations and Dynamic Stabilization
For the time being, we consider only coprime factorizations for systems without direct feedthrough (i.e., D = 0). The formulas for systems with D ^ 0 can be easily constructed from the former ones. The following equations show how coprime factorizations forH(s) = D + C(sI — A)~1Bare constructed from a coprime factorization of G(s) = C(sl — A)~1B:
In the formulas above, the subscript G refers to elements of a coprime factorization of G(s) and the subscript H to those of H(s). These formulas also apply if one wants to construct a coprime factorization of H(s) = D(s) + G(s) in the case that D(s) e H00(£(I7,F)). In the existing theory of coprime factorizations, it is customary to construct coprime factorizations using an exponentially stabilizing feedback operator F and an exponentially detecting injection operator L. In the case of strong stability, this is possible only under extra conditions, which are analogous to those for regular linear systems (see Curtain, Weiss and Weiss [34]). Theorem 6.2.1 Assume that the operators F and L are such that the two systems Y,(A + BF,[B L ] , [ F * C*]*) and E(A + LC,[B L ] , [ F* C* ]*) are input-output stable. Then the following constitutes a doubly coprime factorization ofG(s) = C(sl — A)~1B:
where AF = A + BF and AL = A + LC. Proof It follows directly from the stability assumptions that all components are in H^. That they indeed form a coprime factorization is a matter of algebra, and the computations are the same as in the case of exponential stability (see, for instance, Curtain, Weiss and Weiss [34]). The result above is completely analogous to the corresponding result for regular linear systems in Proposition 3.4 of Curtain, Weiss and Weiss [34]. Their state-space formulas hold under the assumption that ^(A, B, C] is exponentially stabilizable and detectable plus certain regularity conditions on the combination (F, L). These imply the same input-output stability requirements as ours. Staffans has similar results in [98]. Even though our result appears to be nice, there are problems hidden in the assumptions. In contrast to the case of exponentially stabilizable bounded linear systems, in our case it is not at all clear, for a given F and L that are such that E(Ap,B, [ F*,C* ]*) and %(AL, [B L ] ,C) are both input-output stable, that we also have
in HOQ. However, we do need to assume this for the candidate Bezout factors X, Y, X and Y to be in H^. Clearly, this problem does not occur in the case of exponentially
Section 6.2. Coprime Factorizations
71
stabilizable and detectable bounded linear systems. It is exactly this problem that hinders the construction of normalized coprime factorizations for strongly stabilizable and detectable systems later in this chapter. It will also hinder the development in Section 6.3, which deals with designing stabilizing compensators for strongly stabilizable systems. There is one case in which the above mentioned problem does not occur. In the case of a statically stabilizable system, we can choose F — K\C and L = BK^ where both K\ and K-2 are statically stabilizing for Y,(A,B,C), and this will give us a doubly coprime factorization of G(s). Next, we give a different construction, which may lead to coprime factorizations for G(s) — C(sl — A)~1B. Let K(s) be an admissible feedback transfer function for G(s). We define
and
It is a matter of algebra to check that G
and that
So, (6.12), (6.13) define a coprime factorization of G(s] if all the components are in HQQ. We give a number of situations for which this is the case. Lemma6.2.2 LetG(s) = C(sl - A)~1B and let M,N,Y,X,M,N,Y,X be defined as in (6.12), (6.13). 1. If K € HQO w such that (I — GK)~l is a well-posed transfer function and G(I — KG)~l € HOC, then (6.12), (6.13) define a doubly coprime factorization ofG. 2. IfK G HOQ stabilizes G in the input-output sense, then (6.12), (6.13) define a doubly coprime factorization ofG. 3. IfY,(A,B,C) is statically stabilizable, i.e., there exists a K 6 £(Y, U) such that Y>(AK,B,C) is a strongly stable system (where AK = A + BKC), then (6.12), (6.13) define a doubly coprime factorization ofG. In this case, we have the following state-space formulas:
Proof 1. From the assumptions, it follows that G(I — KG) (7-GIT)- 1 e Hoc, because
1
K € HQQ. Consequently,
By assumption, (7 - GK)~1G = G(7 - KG)~l e H^ and so K(I - GK)~1G € H^, which implies via a similar calculation as above that (7 — KG)'1 £ HQQ.
72
Chapter 6. Coprime Factorizations and Dynamic Stabilization
2. This is a direct consequence of part 1, because G(I — KG)~l is one of the components of the closed-loop transfer function Gci (see (6.6)). 3. This is a direct consequence of part 2. In this case, the operator K can be viewed as a (constant) transfer function in HQQ. Because it strongly stabilizes Z(A,B,C), the closed-loop system is certainly input-output stable. The state-space formulas follow by easy calculations from (6.12), (6.13). Note that case 3 is a special instance of case 2, which itself is a special case of 1. Furthermore, note that the factorization for statically stabilizable systems in case 3 is different from the one obtained by taking F = KG, L = BK in Theorem 6.2.1.
6.2.1
State-Space Formulas for Normalized Coprime Factorizations
We are especially interested in normalized doubly coprime factorizations because of their important role in the solution of the robust stabilization problem. They are usually constructed with the use of the stabilizing solutions of the control algebraic Riccati equation and the filter algebraic Riccati equation. In the normalized case, we have to take D directly into account, because our earlier procedure (see (6.8) and (6.9)) to incorporate D in a later stage will not retain the normalization property. Let us therefore consider the control and filter Riccati equations for E(A,B,C,D),
forzi e £>(A) andz 2 eT>(A*). By Theorem 3.3.2 (which can be extended to the 1)^0 case by the same transformation as the one in Section 4.5.4), these equations both have strongly stabilizing solutions if both %(A,B,C) and S(^4*,(7*,S*) are strongly stabilizable and strongly detectable. The corresponding optimal feedback and injection operators are given by
If one takes the same formulas as in Theorem 6.2.1, then, after a normalization, the candidate formulas for the normalized coprime factors would be
where S — I + D*D and R = I + DD*. Because P and Q are strongly stabilizing solutions to their respective Riccati equations, the functions above are all in HQQ. However, this is not at all clear for the candidate Bezout factors
Section 6.2. Coprime Factorizations
73
The problem is the one described below Theorem 6.2.1: we have been unable to show that the terms containing both FQ and Lp are in HQQ. Because P and Q are strongly stabilizing solutions, it does follow, however, that for all u 6 U and y G Y, the functions Y(s)y, Y(s)u, (X(s) - R? }u and (X(s) - 52 )y are in H2. Despite this problem, we have been able to show that (6.18)-(6.21) constitute a normalized doubly coprime factorization for strongly stabilizable and detectable systems with finite-dimensional inputs and outputs under an additional assumption on the spectrum of A. The approach is based on a matrix generalization of the Carleson Corona Theorem. Unfortunately, this approach does not provide us with formulas for the Bezout factors, even though they must exist. Also, the proof does not give any information about whether the functions in (6.22)-(6.25) are in HOQ. This remains an interesting open question. At the end of this section, we show that for systems in the 11(A,B,B*) class with nonnegative, feedthrough operator D, we don't need the assumption of finite-dimensional input and output spaces. For this particular case, we have obtained state-space formulas for Bezout factors. As mentioned, the approach to show that (MQ,NQ) and (Mp,Np) are coprime is based on a matrix generalization of the Carleson Corona Theorem (see Theorem 14-10 in Fuhrmann [46, p. 204] or Theorem 3.1 in [45]). For easy reference, we abstract a special case of the results. Theorem 6.2.3 7. Given M € Hoo(C mxm ) and N G H 00 (C fcxm ), a necessary and sufficient condition for M and N to be coprime over HOQ is the existence of a 5 > 0 such that
where | • || denotes the Euclidean norm. 2. Given M € H 00 (C fcxfc ) and N e H 00 (C /CXm ), a necessary and sufficient condition for M and N to be coprime over HQQ is the existence ofa8>0 such that
We remark that in (6.26), (6.27), Fuhrmann used ||M(s)u|| + ||/V(s)u|| instead of ||M(s)u||2 + ||7V(s)ii||2. Of course, they are equivalent statements, but the version given above is more convenient for our purposes. Using this result, we can give sufficient conditions for the functions in (6.18)-(6.21) to constitute a normalized doubly coprime factorization. Theorem 6.2.4 Suppose that ^,(A,B, C, D) is such that A generates a Co-semigroup on the Hilbert space Z, B e £(C m ,Z), C e C(Z,Ck], D € £(C m ,C fc ). 7. Under the following assumptions, (M.Q,Nq) given by (6.18)-(6.19) constitutes a normalized right-coprime factorization ofG = D + C(sl — A)~1B: (i) A has no essential spectrum on the imaginary axis. (U) There exists an F e £(Z,C m ) such that for all zeZ. (Hi) E(A,B,C, D} is strongly detectable.
74
Chapter 6. Coprime Factorizations and Dynamic Stabilization
2. Under the following assumptions, (Mp,Np) given by (6.20)-(6.21) constitutes a normalized left-coprime factorization of (iv) A* has no essential spectrum on the imaginary axis. (v) There exists an L G £(C fc , Z] such that all z&Z. (vi) E(A*, C*, B*, D*) is strongly detectable. Proof We prove only part 1, as part 2 is its dual. (a) By conditions (ii) and (iii), the control Riccati equation (6.14) has a strongly stabilizing solution which is the unique, self-adjoint, nonnegative solution (this follows from Theorem 3.3.2; see also Remark 3.3.3). Since ApQ is a finite-rank perturbation of A, their essential spectrum is identical (see Theorem 5.35 in Kato [59, p. 244]) and so the essential spectrum of ApQ is not on the imaginary axis. Together with the strong stability of the semigroup generated by AFQ , this implies that ApQ has no spectrum in Re(s) > 0. (b) We now show that MQ and NQ satisfy (6.26). By direct calculation, following Curtain and Zwart [36, p. 372], we find that
where i
. Using the Riccati equation (6.14) and the identities from Curtain and Zwart [36, p. 372] (where , the above simplifies to
and so
which holds for Re(s) > 0 since from part (a) of the proof, there holds
In particular,
whence Recall that the candidate Bezout factors X and Y in (6.22), (6.23) are in H2 (modulo a constant) and so the shifted factors X(s — e), Y(s — E) are in HQQ for each E > 0. In effect, we have a coprime factorization on Re(s) > e,
with MQ, NQ, X, Y all holomorphic and bounded on Re(s) > e. Applying Theorem 6.2.3 to this situation, we conclude that there exists a S(e] > 0 such that
Section 6.2. Coprime Factorizations
75
We need to show that 6(e) has a nonzero lower limit as £ tends to zero. Suppose, on the contrary, that there exists a sequence un G Cm, \\un\\ = I and zn € Cj such that ||-^Q(2n)wn|| < 1/n and ||NQ(zn)un|| < 1/n. Since un is a bounded sequence in Cm, it has a convergent subsequence unk —> £ as k —> oo, and we have
for sufficiently large k. Similarly, ||-/VQ(znfc)£|| < 2/rifc. Now, (6.30) with u = £ implies that for sufficiently large n^, Re(,znjt) < £. Since £ can be chosen arbitrarily small and (6.29) with u = £ holds on the imaginary axis, we can conclude that there exists no such sequence zn in CQ". (c) So, we have shown that (MQ,NQ) represents a normalized right-coprime factorization for G. We end this section by giving two important corollaries to this theorem. Corollary 6.2.5 Consider a system %(A,B,C, D) with finite-dimensional input and output spaces. Under the conditions (i)-(vi) (see Theorem 6.2.4), this system has a normalized doubly coprime factorization G — NqM^1 = MplNp, with coprime factors given by(6.18)-(6.21). Proof This follows directly from Theorem 6.2.4 above and Lemma A.7.44 in Curtain andZwart[36]. A more balanced, slightly stronger version of Theorem 6.2.4 is the following corollary. Corollary 6.2.6 Suppose that Y,(A,B,C, D) is such that A generates a Co-semigroup on the Hilbert space Z, B € £(C m ,Z), C e £(Z,C fc ) and D € £(C m ,C fc ). If A has compact resolvent and £(A B, C, D} is strongly stabilizable and strongly detectable, then (MQ,NQ) and (Mp,Np) given by (6.18)-(6.21) are normalized right-coprime and leftcoprime factorizations ofG(s], respectively. Proof If A has compact resolvent, then (i) and (iv) hold and A + BF generates a strongly stable semigroup if and only if A* + F*B* does. A consequence of the above theorem is that there exist Bezout factors X, y, X, Y 6 HQQ satisfying (6.5). Note that Bezout factors are not unique and we have not shown that the candidate Bezout factors X,Y,X,Y in (6.22)-(6.25) are true Bezout factors. We know only that they are in H2 (modulo a constant), but not that they are in H^. While this is not essential for the robust stabilization problem with respect to coprime factor perturbations (which we address in Chapter 7), it is needed for dynamic stabilization results, which are the topic of the next section. This remains an interesting open question. It is interesting to note that the example in Section 9.4, a model describing the dynamics of a flexible structure, does satisfy the assumptions of Theorem 6.2.4. Hence, it has a normalized doubly coprime factorization given by (6.18)-(6.21). Another important class of systems satisfying the conditions of Corollary 6.2.6 is that of systems in the E(^4, .£?,£*) class with finite-dimensional input space U. We can relieve the assumption on the dimension of U for systems in the E(A,B,B*) class with a nonnegative feedthrough operator D. The remainder of this section will be devoted to this subject. The approach is to construct explicitly Bezout factors for this class. We need the following result, which is taken from Curtain and van Keulen [33].
76
Chapter 6. Coprime Factorizations and Dynamic Stabilization Lemma 6.2.7 IfG(s) is positive real, then
form a doubly coprime factorization ofG. The following result is very close to Theorem 12 in Curtain and Zwart [37]. They needed to restrict their theorem to a finite-dimensional input space. Theorem 6.2.8 Consider the system E(A,B,B*,D) under assumptions A1-A2 (see page 17). Assume, furthermore, that D = D* > 0. Then the following define a normalized doubly coprime factorization ofG(s) = D + B*(sl — A)~1B:
where S = I + D2, FQ =-S~^(B*Q +DB*}, LP = -(PB + BD)S~1, AFQ =A + BFQ, AI,P = A + LpB*. P and Q are the minimal self-adjoint, nonnegative solutions of the Riccati equations
for zi e T>(A) and z2 € T>(A*). Proof Our assumptions ensure that S(A, B, B*, D) satisfies properties P4 and P5 from Lemma 2.2.6. Thus, we can apply Theorem 3.3.1 (see also Remark 3.3.3) to show that each Riccati equation has a minimal self-adjoint, nonnegative solution, Q and P, respectively. Moreover, Theorem 3.3.1 shows that FQ and Lp are such that their corresponding closedloop systems are input-output stable. Consequently, Mp, Np, MQ and NQ € HQQ. It is a matter of long but straightforward computation to show that MQ , NQ , Mp, Np ,Y,X,Y,X satisfy (6.1)-(6.5). For the normalization properties (6.2) and (6.4), these computations are the same as the corresponding ones in Theorem 7.3.11 and Exercise 7.29 in Curtain and Zwart [36] which make use of the fact that P and Q satisfy their respective Riccati equations. To show that the given factorization is indeed a coprime factorization, we need to show thatX,y,X,yGHoo(£(tf)). We will do this only for X(s) = (NQ+MQ}-1, as the proof for X(s) is analogous. Now,
so,
Section 6.3. Dynamic Stabilization
77
Because D > 0 implies that (/ + D) ~l > el for some e > 0, we can factor (/ + D] ~l = LL*, with L boundedly invertible. Now, the system E,(A,BL,L*B*) satisfies assumptions Al and A2, because Y,(A,B,B*) does so. Hence, property P5 of Lemma 2.2.6 applies, and it follows that E(A - BLL*B*,BL,L*B*} is input stable, i.e.,
where ABL = A-BLL*B = A - B(I + D)~1B*. By the invertibility of L, we obtain oTa\\zeZ. Consequently,
Next, we show that
To do so, note that
Therefore, MQ(s)X(s) = (I + G(s))-1 and NQ(s)X(s) = G(s)(I+ G(s))-\ Because G(s) is positive real, we can apply Lemma 6.2.7 to show that MQ(S)X(S) and NQ(S}X(S] e H 00 (£(C/)). From the normalization of the factorization, we obtain
By the above observation, the right-hand side is in L 00 (— j'oo, joo; £(^0) anc^ so me same holds for the left-hand side. Thus, X~(jw} G LOO(— joo,joo; C(U}}. Combining this with (6.31), we obtain that X ( s ) € H 00 (£(C7)), by Lemma 4.2.4.
6.3 Dynamic Stabilization In this section, we investigate the feasibility of the use of observer-based dynamic compensators for stabilization of strongly stabilizable systems. We restrict ourselves to systems without direct feedthrough (i.e., the case D = 0). The extension to the case that D ^ 0 is straightforward and comes only at the cost of more complicated formulas. The setup that we consider is the classical one and it is schematically represented in Figure 6.2.
Figure 6.2: Configuration for observer-based control.
78
Chapter 6. Coprime Factorizations and Dynamic Stabilization
Writing down the dynamics in terms of the plant-state z and the state-estimation error e := z — z, we obtain the equations
In what folllows, we denote Ap = A + BF and AL = A + LC. Mimicking the case of exponentially stabilizable systems, we take F and L to be a strongly stabilizing feedback operator and a strongly detecting injection operator, respectively. More precisely, we assume that
We stress that assumptions SI and S2 are stronger than just strong stabilizability or strong detectability. The difference is that in S1 we also demand input-output stability and input stability and that in S2 we also demand input-output stability and output stability. The strong connection between coprime factorizations and dynamic stabilization is well known and was described in Section 6.1. Indeed, problems similar to the ones that were found in the previous section arise here. We start by turning our attention to the question of what compensators of the form sketched above do achieve input-output stability of the closed-loop system. Later in this section, we consider strong stability of the closed-loop system. The closed-loop transfer function is given by
where
If we describe the closed-loop dynamics in terms of z and e instead of z and e, we obtain a different representation of this transfer function:
Section 6.3. Dynamic Stabilization
79
We now have the following theorem. Theorem 6.3.1 Let F e £(Z, U) and L € C(Y, Z). Consider the conditions is input-output stable, and | is input-output stable. If either SI and S3 or S2 and S4 hold, then the compensator
stabilizes Y,(A,B,C} in the input-output sense. Proof We show that the combination S1 and S3 is a sufficient condition for stabilization by appealing to (6.32). That S2 and S4 are sufficient as well follows completely analogously by using the alternative representation (6.33) of the closed-loop transfer function. By SI, the transfer functions C(sI-AF)~1B, F(sI-AF)~lB are in H^ and by S3, F(sl - AL)~IB e HOC and F(sI-AL)~1Le H^. This implies that all components of Gci are in HQO, and so the closed-loop system is input-output stable. D It follows from the proof of the above theorem that SI and S2 are unnecessarily strong assumptions for this result. It would be sufficient to demand that the systems in S1 and S2 are input-output stable. Next, we turn our attention to conditions under which application of a compensator does lead to a strongly stable closed-loop system. Clearly, the assumptions of Theorem 6.3.1 will be necessary here because input-output stability is weaker than strong stability. In fact, the conditions that we need in this case are similar to the conditions in the previous theorem. Theorem 6.3.2 Let F € C(Z, U) and L € £(Y, Z). Consider the conditions is a strongly stable system, and is a strongly stable system. If either SI and S5 or S2 and S6 hold, then the compensator (6.34) makes the closed-loop configuration of Figure 6.2 a strongly stable system. Proof We show that the combination of S1 and S5 is sufficient for strong stabilization by appealing to (6.32). The proof that S2 and S6 are sufficient as well is completely analogous but uses (6.33) instead. First, we prove the strong stability of the closed-loop semigroup. That is, we show that (z(t),e(t)} converge to zero as t tends to infinity. These variables satisfy
80
Chapter 6. Coprime Factorizations and Dynamic Stabilization
where TL (t) and TF (t) are the semigroups generated by A + LC and A + BF, respectively. By S5, TL (t) is a strongly stable semigroup and so e(t) —> 0 as t —> oo. To show that z (t) —> 0 as t —> oo, we want to apply Lemma 2.1.3. To do so, we need to show that TF(t) is a strongly stable semigroup, E(AF,B, —) is input stable and that FTi,(-}eQ G L 2 (0,oo;t/) for all eo G Z. The first two demands are satisfied by S1; the last one is equivalent to the output stability of S(A/,, —,F), which is guaranteed by assumption S5. As S5 implies S3, Theorem 6.3.1 applies to show that the closed-loop system is inputoutput stable. For output stability of the closed-loop system, we need to show that the following transfer function is in H 2 (t/ x Y) for all z\,Z2 G Z:
Sufficient for this to hold are that H 2 (t/) for all z G Z. The last condition follows from S5 and the others from SI. Finally, input stability of the closed-loop system means that for all z\, 22 G Z,
is in U2(U x Y). This is the case if HOC (£([/)). The first condition is satisfied by S1, the others by S5. As in the case of the formulas for coprime factorizations in the previous section, the conditions that we obtained are somewhat problematic. For given F and L satisfying SI and S2, it is in general not clear whether they also satisfy one of the conditions S3-S6. In particular, it is not clear whether any of these conditions are satisfied for the LQG feedback and injection operators FQ and Lp. This case suffers from exactly the same problem as the result on normalized doubly coprime factorizations: It is not clear if C(sl - AFQ)~1LP, FQ(sI - ALp)~1B and FQ(sI - ALp}~lLP are in HOO. Note that the situation here is even somewhat worse than in Section 6.2. We would need to know that A + LpC generates a strongly stable semigroup, but the Riccati theory only allows us to conclude that its adjoint generates a strongly stable semigroup. These problems make the present results not completely satisfactory. For statically stabilizable systems, we give a number of special cases in which the conditions are satisfied. • Take F = KiC and L = BK2, where for i = 1,2, E(4 + BKiC, B, C) is a strongly stable system. Then all of the above conditions S1-S6 are satisfied and so the corresponding compensator yields a strongly stable closed-loop system. • Let L be any strongly detecting injection (i.e., such that S2 is satisfied) and choose F = KC, where K is statically stabilizing E(^4,B,C). Then the corresponding compensator is strongly stabilizing, because conditions SI, S2 and S5 are satisfied. • Let F be any strongly stabilizing feedback (for instance, F = FQ) and choose L = BK, where K is statically stabilizing E(^4,S,C). Then the corresponding compensator is strongly stabilizing, as assumptions SI, S2 and S6 hold.
Section 6.3. Dynamic Stabilization
81
Very often, dynamic controllers are designed that render the closed-loop system inputoutput stable. A natural question is then whether the resulting closed-loop system is also strongly stable. In Theorem 6.3.2, we showed how to construct a strongly stabilizing compensator. The results that follow now give different conditions under which an input-output stabilizing compensator renders the closed-loop system strongly stable. These results are particularly applicable (but certainly not restricted) to the situation when a strongly stabilizable and detectable system is stabilized in the input-output sense by a finite-dimensional controller. In the remainder of this section, we do not assume that the direct feedthrough operator D = 0, in order to give an idea of how the formulas change in this case. Let us consider the configuration of Figure 6.1, and let the systems EG and SK have the state-space realizations E(AG,BG,CG,DG} and Y,(AK,BK,CK,DK}, respectively. Theorem 6.3.3 Consider the feedback interconnection as in Figure 6.1 of the systems ^(AG,BG,CG,DG} and %(AK,BK,CK, DK} with transfer functions G(s) and K(s} and state spaces ZG and ZK, respectively. Assume that K(s] is an admissible feedback transfer function for G(s). Let the closed-loop system be described by the state-space realization Y,(Aci,Bci,Cci,Dci} on the product state space ZG x ZK and let Gci denote the closed-loop transfer function. 1. If E(AG,BG,CG,DG) and E(AK,BK,CK,DK) are both strongly stabilizable, then *£i(Aci, Bci, Cci, Dc\) is strongly stabilizable, as well. 2. If E(AG,BG,CG,DG} and Y,(AK,BK,CK,DK} are both strongly detectable, then ^(Aci, Bci, Cci, Dc{) is strongly detectable, as well. 3. If II(AG,BG,CG,DG) and Y,(AK,BK,CK,DK) are both strongly stabilizable and strongly detectable and Gci € H^, then H(Aci, Bci, Cci, Dc{) is a strongly stable system. Proof It can be verified by straightforward calculations that the closed-loop system E(Ad, Bd,Cd,Dd) is given by
1. By the stabilizability assumption, there exist FG € £(Zc,U) and FK € £(Zx,Y) suchthatX(AG + BGFG,Q,[FG CG]*)andE(AK + BKFK,0,[F*K C*K ]*) are strongly stable systems. It can now easily be verified that with
82
Chapter 6. Coprime Factorizations and Dynamic Stabilization
we obtain
and ^(Aci + JE?c/.Fcj,0, [ F*j C*t ]*) is a strongly stable system. Hence, the closed-loop system E(^4 c /,B c i,C c /,Z) c i) is strongly stabilizable. 2. By assumption, there exist operators LG € £(Y,Zc) and LK 6 C(U,ZK] such that E(4G 4- £ G Cc, [ LG BG ] ,0) and S(4K + L/C CK, [ £* BK ] ,0) are strongly stable systems. It can now easily be verified that with
we obtain
and %(Aci + LciCci, [ Lc\ Bc\ ] ,0) is a strongly stable system. Hence, the closed-loop system Y,(Aci,Bci,Cci,Dci) is strongly detectable. 3. By parts 1 and 2 of this theorem, the resulting closed-loop system is strongly stabilizable and detectable. By assumption, it is input-output stable. Hence, we can apply Theorem 3.2.2 to prove that the closed-loop system is strongly stable. In the literature, there are many results on stabilizing infinite-dimensional systems in an input-output sense by rational controllers which are usually (strongly) stabilizable and detectable. Theorem 6.3.3 gives testable conditions for internal asymptotic stability of the resulting closed-loop system. We also have a version of the above theorem for the statically stabilizable case. Theorem 6.3.4 Suppose that the systems EG and E/<- ore connected in feedback as in Figure 6.1 and that I — DG&K is boundedly invertible. If both EG and EX are statically stabilizable, then the closed-loop system !Z(Aci,Bci,Cci,Dci) is statically stabilizable as well. If, in addition, the closed-loop system is input-output stable, then it is a strongly stable system. Proof Analogous to the proof of the previous theorem, it is easy to construct an output feedback operator that statically stabilizes the closed-loop system from the output feedback operators that statically stabilize the two subsystems EG and S/f. The assertion about the strong stability of the closed-loop system follows from part 3 of Theorem 6.3.3 in combination with the fact that static stabilizability implies strong stabilizability and strong detectability. As an application of these results we study the stability of the feedback interconnection of a system in the E(A,B,B*} class with a finite-dimensional controller. In the literature on flexible structures, one does consider strong stabilization (in the sense of obtaining a strongly stable closed-loop semigroup) by finite-dimensional strictly positive real dynamic compensators. Examples of this can be found in a number of papers by Morgiil [75,76,77]. In these papers, ad-hoc methods are used to prove strong stability of the semigroup. The following result shows that in this case, the resulting closed-loop system is in fact a strongly stable system.
Section 6.3. Dynamic Stabilization
83
Figure 6.3: Negative feedback connection of E(A,B,B*,D) with a finite-dimensional strictly positive real compensator. Theorem 6.3.5 Consider the system 2,(A,B,B*,D] under assumptions A1-A5 oj'Section 2.2, connected in feedback with a finite-dimensional compensator ^(Af, B f , Cf, Df) as in Figure 6.3. Assume, furthermore, that 1. the input space U is finite-dimensional, 2. D + D*>0, 3.
Ff is strictly positve real ,and
4. Y,(Af,Bf,Cf,C Df) i5 stabilizable and detectable. Under these conditions, the closed-loop system is a strongly stable system. Proof It is a well-known result that the negative feedback interconnection of a positive real system with a strictly positive real system is input-output stable (see, for instance, Desoer and Vidyasagar [39]). Now, E(A,B,B*,D) is statically stabilizable and Jl(Af,Bf,Cf,Df)Dis stabilizable and detectable. Hence, part 3 of Theorem 6.3.3 applies to show that the closed-loop system is a strongly stable system. D This result thus states that the interconnection of a system in the E(A,B,B*) class with a strictly positive real finite-dimensional stabilizable and detectable system leads to a strongly stable system. This is a very nice extension of the well-known fact that a system in the E(^4, £?,£?*) class can be made strongly stable by a static negative definite output feedback. Related results on input-output stabilization by dynamic compensators for a class of infinite-dimensional models for flexible structures can be found in Balakrishnan [7, 8, 9].
This page intentionally left blank
Chapter 7
Robust Stabilization In this chapter, we consider the problem of robust stabilization of a strongly stabilizable linear system with respect to coprime factor perturbations. The configuration for stabilization that we consider is the one from Chapter 6: the plant and a dynamic compensator are connected in feedback as in Figure 6.1 and we aim at input-output stability of the closedloop system. The controller, however, is not expected to stabilize just the plant: it should stabilize all plants which are close to the plant in the gap metric. That is, if G = M~1N is a left-coprime factorization, it should also stabilize all plants
for any [ AW AM ] € HQQ having Hoc-norm less than some e. Such a controller is called a robustly stabilizing controller with robustness margin e. The problem is to find the maximal s for which there exists such a controller and to find all controllers which stabilize G with a robustness margin at least equal to e. A precise formulation of this problem will be given in Section 7.2 In both the finite-dimensional theory and the theory of exponentially stabilizable infinitedimensional systems, the problem of robust stabilization with respect to coprime factor perturbations is reduced to a suboptimal Nehari problem and the parameterization of all robustly stabilizing controllers is derived from the parameterization of all solutions to the Nehari problem. We use the same approach and so we devote the next section to a treatment of the Nehari problem associated with nonexponentially stabilizable systems.
7.1
The Nehari Problem
The Nehari problem has received a lot of attention, both in the mathematical and the engineering literature (see Adamjan, Arov and Krein [1], Ball and Helton [10], Treil [102]). It has entered the control literature due to its connections with the parameterization of H^ controllers (see, for instance, Green et al. [52]). For an operator-valued function G e L 00 (£(t/,y)), the Nehari problem is to find the minimum distance of G from the space of antistable operator-valued functions, i.e.,
where U and Y are separable Hilbert spaces. It is well known that the infimum equals the norm of the Hankel operator with symbol G, \\Hc\\, which will be defined later in this 55
86
Chapter 7. Robust Stabilization
section. (See Power [87], Partington [84] or Treil [102] for details on Hankel operators and the solution to the Nehari problem.) The suboptimal Nehari extension problem is to find all K(—s) € HQQ (£(£/, F)) that satisfy
for a given a > \\Hc\\- This problem has been solved in the literature for any matrix-valued function G € L^ (see Adamjan, Arov and Krein [1], Ball and Helton [10], Treil [102]). The solution in the frequency domain for \\Hc || < 1 is given in terms of a linear-fractional parameter O = [ ©n 612 ©21 ©22 ] with entries possibly only in H^. Moreover, this parameter arises from the J-spectral factorization W = X~JX for
For control applications one needs to give a state-space interpretation for the solutions. One supposes that G(s) has a realization G(s) = C(sl — A)~1B, where A,B,C are the generating operators of the state-space system, and one tries to find formulas for the solutions K in terms of these generating operators. These solutions then provide the parameterization of all stabilizing controllers for the H^ control problem. This has been carried out in Green et al. [52] for the case that A, B, C are matrices. This problem has also been studied in the literature, assuming that A is the generator of an exponentially stable Co-semigroup and that B and C are linear operators. In Curtain and Zwart [36], U and Y are assumed to be finite-dimensional and B and C are bounded operators. Extensions to cases in which U and Y are infinite-dimensional or B and C are unbounded or both can be found in Glover, Curtain and Partington [50], Ran [88], Curtain and Ran [32], Foias and Tannenbaum [44], Curtain and Zwart [35] and Curtain and Ichikawa [25]. In all these papers, the assumption is made that A generates an exponentially stable Co-semigroup and the Hankel operator is compact. However, we will study this problem for the case that 1£(A,B,C) is not an exponentially stable system. Unlike in other chapters of this book, here we do not need to assume that Yi(A,B,C) is strongly stable or strongly stabilizable. The specific class of systems we consider in this section are the systems E(A,B,C) that satisfy the following hypotheses: HI. A generates a Co-semigroup on the separable Hilbert space Z and B € £(U,Z), C € £(Z,Y), where U and Y arefinite-dimensionalHilbert spaces;
is input stable; is output stable. We remark that, instead of H2, we could have assumed that G(s) G Hoo(£(t/,y)) (i.e., S(A,S,C) is input-output stable) because, by Lemma 4.2.4, the combination of input stability with G G L^C^Y)) implies that G G R^jC^Y)). Theorem 3.1 in Hansen and Weiss [53] (also see our Lemma 2.1.4) states that the system is input stable if and only if the Lyapunov equation
Section 7.1. The Nehari Problem
87
has a self-adjoint, nonnegative solution (see also Grabowski [51]). In this case, one such solution is the controllability Gramian LB of S(.A, B, C}, which is defined as LB = $$*, where $ is the extended input map (see (2.10)). By duality, the system is output stable if and only if the Lyapunov equation
has a self-adjoint, nonnegative solution. In this case, the observability Gramian LC of S(j4,jE?,<7), which is defined as LC = *&*$, where \I> is the extended output map, is one such self-adjoint, nonnegative solution of (7.5). We remark that for the theory in this section, we do not require that the Lyapunov equations (7.4) and (7.5) have unique solutions; this will be the case if A* (respectively, A) generates a strongly stable Co-semigroup (see Lemma 2.1.4). However, we prefer not to make this assumption as it is unnecessary for our goal, which is to solve the Nehari problem (7.1) and its suboptimal extension in terms of the parameters A, £?, C. The main results of this section are the following two theorems. Theorem 7.1.1 Consider G(s) = C(sl - A)~1B under assumptions H1-H4. The following three statements are equivalent. 1. There exists a K(-s) € H00(£(C/,y)) such that
2. a > \\HG\\ — 7"2 (LB^C), where r denotes the spectral radius. 3. There exists an X(-s) € H 2 (£(F x £/)) such that X'^-s) e H2(C(Y x U)} and X^(-s) e HOC (£(t/)), which satisfies the factorization W = X~JX, where W and J are given by (7.3). Theorem 7.1.2 Let the assumptions H1—H4 be satisfied and let a > r*i (L^Lc). All solutions K(—s) e HQQ(£(U, Y)) to the suboptimal Nehari extension problem (7.6) are given by
where
andQ(-s) e Hoo(£(C7,y)) satisfies \\Q\\n < 1. To prove these theorems we have to develop some intermediate results. First we study the Hankel operator associated with our system. Hankel operators are at the core of the solution to the Nehari problem and we need to make the link between the frequency-domain Hankel operator and the time-domain Hankel operator. For a transfer function G(s) — C(sl — A)~1B e Loo(£(t/,y)), the frequency-domain Hankel operator HG : H 2 (t/) -> H2(F) is defined by
88
Chapter 7. Robust Stabilization
for / e H 2 (C/), where /_(s) := /(—s), AG is the multiplication map induced by G and TT is the orthogonal projection from L 2 (— joo, joo; V) onto H2(y). The time-domain Hankel operator Y :L 2 (0,oo;f7) —>• 1,2(0,00; F) is defined for / € L2(0,oo;£7) by
where we assume that the impulse response h(-} = CT(-}B € L2(0,oo;£(t/,y)). This assumption is not sufficient to ensure that F is bounded, nor that F is isomorphic to HGWe quote sufficient conditions from van Keulen [103]. Lemma 7.1.3 Suppose that h € L<2(0, oo;£(t/, Y)) and h G LQQ (C(U, Y)), where h is the Laplace transform ofh. Then, F G £(L,2(0,oo; £/),L2(0,oo; Y")) one?
for f G L2(0,oo;t/), where again ~ denotes the Laplace transform. Moreover, h G H00(£(f/,F)). The crucial condition in this result is that h G L^; in the absence of this condition it can happen that the time-domain Hankel operator F corresponds to a symbol different from h (see [103]). We state the following results without proof. Some of them are easy to prove, the others are well known. Lemma 7.1.4 1. Let f ( s ) = C(sI-A)~lB. f is holomorphic on Cj if and only if /~(s) := [/(—s)]* = B*(—si — A*)~1C* is holomorphic on CQ. f ( s ) is holomorphic on Cj if and only if g(s) := /~(-s) = B*(sl - ^l*)-1^ is holomorphic on Cj. Moreover, f € HOO if and only ifg € HQQ and in this case \\f\\oo — \\9\\oo, and f G H2 if and only ifg e H2 and in this case \\f\\2 = IMh2. If C(sl - A) ~1 Bu e H2 (Y)for allu^U and U and Y arefinite-dimensional,then C(sI-A)-lB£H2(jC(U,Y)). 3. Iff € Hoo(£(Z)) andg e H 2 (£(Z)), then fg G H 2 (£(Z)) am/«// € H 2 (£(Z)). Next, we prove a sequence of lemmas under the assumptions H1-H4. First, we apply Lemma 7.1.3 to our class of systems and make a link between the Hankel operators, the extended input and extended output maps and the controllability and observability Gramians, as introduced in (7.4) and (7.5). Lemma 7.1.5 Consider E(A,B,C) under the assumptions H1-H4, let h = CT(-}B be its impulse response and let G(s] — C(sl — A)~1B be its transfer function. is isomorphic to
where r denotes the spectral radius.
and
Section 7.1. The Nehari Problem
89
Proof 1.
by the semigroup property
is bounded. Thus, for all by H3 and H4 and G(s) € LOO by H2 (see Chapter 4 in [36] for h(s) = G(s}\ So, Lemma 7.1.3 applies to show that F is isomorphic to HQ and G e HQQ(£(U, V)). Finally, since is self-adjoint see Lemma A.4.15 in [36] 2. Note that implies that the spectral radius of is less than 1. Hence, is boundedly invertible. Next, we derive some results on two Riccati equations, which we need for the (crucial) Lemma 7.1.7. Lemma 7.1.6 Suppose that >
1.
is a nonnegative solution of the following Riccati equation for
where
generates the Co-semigroup Tw(t).
2.
is a nonnegative solution in £(Z) of the following Riccati equation for z € T}(A):
where
generates the Co-semigroup Tx(t).
3. For all
90
Chapter 7. Robust Stabilization
Proof Parts 1 and 2. First, we need to show that WT>(A*) C T>(A) and XT>(A) C T>(A*}. The proofs of Lemmas 4.1.24 and 8.3.2 in [36] apply here to show that LBV(A*} C -D(A), LCT>(A) c T>(A*), NaV(A) C T>(A) and N;T>(A*) C T>(A*). That W and X satisfy (7.11) and (7.13), respectively, is then a matter of algebra using the Lyapunov equations (7.4) and (7.5). To show that W > 0, consider
The proof that X > 0 is analogous. 3. We prove only the W-results, as those for X are analogous. Appealing to the PaleyWiener theorem (see, e.g., Theorem A.6.21 in [36]), it is sufficient to derive the time-domain inequalities
These follow from (7.11) substituting z = T^(t}z\ for z\ e T>(A*} and taking inner products with z to obtain
Integrating from 0 to t yields
Since W > 0 and T>(A*} is dense in Z, this proves (7.15). Most of the recent papers on the suboptimal Nehari extension problem exploit the relation with a certain J-spectral factorization. We will do the same and in the following lemma we give a condition for the existence of the factorization and derive a number of properties of the spectral factor. Lemma 7.1.7 Let G(s) = C(sl - A)~1B, and assume that conditions H1-H4 are satisfied. Let a > r? (LsLc) and define X(s) by
X has the following properties.
Section 7.1. The Nehari Problem
91
1. X satisfies the factorization for u G R,
where
2. X(s) is invertible and its inverse V(s) is given by
5. Xn(—s) is invertible and its inverse is given by
6. V<32(—s) is invertible and its inverse is given by
and is given by
and is given by
Proof The proofs of all the explicit formulas follow by algebraic manipulations using the Lyapunov identities (7.4), (7.5) and the Riccati equations (7.11), (7.13), noting that LBT>(A*} C T>(A), etc. For details see [36, Corollary 7.3.7 and Lemma 8.3.3]. As an example, we calculate
92
Chapter 7. Robust Stabilization
where we use Corollary 7.3.7 from [36] for the second equality. Then,
The formulas for X111, X^iX^ and XnlXi2 can be calculated in a similar fashion, yielding the identities asserted in parts 7 and 8. The tricky part is to show that the transfer functions are in the right spaces. From H4 we obtain that Xn(-s) -1 € H2(£(Y)) and X2i(-s] € H2(C(Y,U}}. It follows from H3 that V21(-s) € H 2 (£(r,t/)) and V22(-s) - ±1 e H 2 (£(T/)). For this, we use part 2 of Lemma 7.1.6. Using part 3 of Lemma 7.1.6, we deduce that V r 2 2 1 (—s] — al € H2(£(t/)), x ii(-s) -1e H 2(£(F)), and the functions in parts 7 and 8 are in H2(jC,(Y,U}) and H 2 (£(t/,y)), respectively. To prove that the functions in parts 5, 6, 7 and 8 are also in H^, we first deduce some inequalities from the factorization (7.17) and its inverse relation
The (1,1)-block of (7.17) yields
whence,
The (2,2)-block of (7.19) yields
and
Section 7.1. The Nehari Problem whence,
93 , and
Now, Lemma 4.2.4 applies to show that X^l(-s) e Hoo(£(y)), V^l(-s) & Hoo(£(E/)) and the functions in part 7 are in H 00 (£(y,U}}. To show that V 1 2 (—s)V^ l (—s) = X^(-s)X12(-s) e H 00 (£(Z/,y)), we introduce K(-s) = Vl2(-s}V^(-s}. Then,
and for s = juj, u> € R, we have
where we have used the spectral factorization. Thus, G + K € L^ and ||G? + .K"||oo < cr. Consequently, K € LQQ and ||.K"||oo < Halloo +°" < °o. Previously, we proved that K(-s) € H 2 (£(l/,y)). So, Lemma4.2.4 applies to show that #(-s) € H 00 (£(f/,y)). Finally, we must show that X i 2 ( — s ) , X^—s), V\\(—s) and Vi 2 (—s) G H 2 . From the identity XV = VX - /, we obtain
Furthermore,
The results now follow, using part 3 of Lemma 7.1.4 and the fact that the terms between curly braces are in H2 n H^ and the others in FLj. Although the above proof shows many similarities to proofs in Curtain and Zwart [35, 36], there is a significant difference. The exponential stability assumption made in [35, 36], means that it is fairly easy to show that X(—s) and V(—s) are in Hoo(£(y x U}}. In our case, however, this assumption no longer holds and we obtain only V(—s), X ( — s ) € H2(£(y x U)) and that some components are in HOQ. Fortunately, these are the crucial ones we need. Because X and V are not in HOC, the factorization in (7.17) is not a J-spectral factorization, according to the usual definitions. One would expect that the above results are also valid even if U and Y are infinitedimensional. However, in this case the situation becomes very complicated. We were able to prove that V(—s)(u,y) e H2(y x U) for all u e U and y E Y, but we could not prove a similar claim for X ( — s ) . The main reason for this is that part 2 of Lemma 7.1.4 is not valid in this case, and that the second claim in part 3 of Lemma 7.1.4 does not extend to functions g(s) that satisfy only g(s)z e ^(Z) for all z e Z.
94
Chapter 7. Robust Stabilization Now we are able to prove our main results. Proof of Theorem 7.1.1 1 =» 2. It is well known (see Treil [102]) that
In Lemma 7.1.5, it was shown that Combining the two, we obtain statement 2. 2 => 3. We have shown this in Lemma 7.1.7. 3 =$> 1. We take K(-s) = Vn(-s)V^l(-s). In the proof of Lemma 7.1.7, this K was shown to satisfy K(-s) 6 H 00 (£(t/,y)) and ||G + tf||oo «r. Proof of Theorem 7.1.2 Suppose that K(—s) is given by (7.7) and (7.8) for some Q € Hoo(C(U,Y)), IIQHoo < 1. In the proof of Lemma7.1.7, we showed that V^V2i satisfies HV^VbiHoo < 1. Hence, H^V^QIU < 1 and (I+ V221V21Q)~1 <= ^(£(£7)), because HQO is a Banach algebra. Consequently,
In addition, by part 4 of Lemma 7.1.7 and part 3 of Lemma 7.1.4,
So,
is the sum of an H2- and an Hoo-function, because (Ri(—s) — Q(—s)) € H2(£(t/)) and both ^? 2 ~ 1 (~ 5 ) and Q(~s) € Hoo(£(?7)). We can show analogously to the derivation in (7.22) that on the imaginary axis,
and so G + K € L^ and, consequently, K € LQQ . Moreover, this inequality shows that all K given by (7.7) and (7.8) satisfy ||G + .fiT|| < a. Lemma 4.2.4 applies to show that
tfC-^eHooC/Xtf.y)).
For the converse, suppose that K(—s) G Hoo(£(t/,F)) satisfies (7.2). We seek a representation K(—s) = -Ri(—s)R2(—s)~ l . Define
From part 3 of Lemma 7.1.4 and part 3 of Lemma 7.1.7, we deduce that both U\(—s) and 1/2 (—s] are sums of H2- and HQQ-functions. As in the proof of Theorem 8.3.7 of [36], we can deduce the following results:
Section 7.1. The Nehari Problem
95
U<2.(ju} is invertible over £>(U) for all u; € R and
satisfies
We now identify #2(—s) = U2 1(—s), RI(—S) — K(—s)R2(—s), which give an appropri ate representation,
It remains to prove that U(-s) € Hoo(£(E/,y)). Define
F is in Loo(£(t/)) by (7.21) and (7.26) and UF^ < 1. Because Loo(£(t/)) is a Banach algebra, we can conclude that / — F is invertible over L^ and using the representation (7.23), it is easily checked that
Thus,
and (7.24) shows that C/i(—s) € L^ (£(£/, F)), as well. We have already proved that U\(—s) and Uz(—s) are the sums of H2- and Hoo-functions and so by Lemma 4.2.4, we conclude that they are both in HQQ . Thus,
is the sum of a function in H2 and a function in HOC. This, together with (I — F ( s ) } 1 €Loo, shows that it is also in HQO, again using Lemma 4.2.4. By Lemma 8.3.5 of Curtain and Zwart [36], we obtain that F(s) € H 00 (£(C/)). As V 22 (-s) -a~ll e H 2 (£(C/)),
is the sum of a function in H2 and a function in HOQ, and as U\ G HQQ,
is the sum of a function in H^ and a function in HQQ, as well. However, we have already proved that U(—s] G L^(£([/, Y}} and so we can apply Lemma 4.2.4 to obtain that
96
Chapter 7. Robust Stabilization
7.2 Parameterization of Robustly Stabilizing Controllers In this section we formulate the problem of robust stabilization and we present a parameterization of controllers that solve the problem. We consider the robust stabilization problem for those systems for which we obtained explicit formulas for a normalized doubly coprime factorization in Chapter 6. More explicitly, we consider systems £(-A, B, C, D) with finitedimensional input and output spaces that satisfy assumptions (iMvO of Theorem 6.2.4: (i) A has no essential spectrum on the imaginary axis; (ii) there exists an Fe£(Z,C w ) such that all z e Z; (iii) E(A,B,C,D) is strongly detectable; (iv) A* has no essential spectrum on the imaginary axis; (v) there exists an L e £(Ck,Z) such that [ B L ]* (sI-A^^z € H 2 (C7x Y) for all z&Z; (vi) E(A*,C*,B*,D*) is strongly detectable. A class of systems satisfying these assumptions is that of strongly stabilizable and detectable systems E(A,B,C,D), with finite-dimensional inputs and outputs and where A has compact resolvent. An important special case is that of systems in the S(A, £?,£?*) class with finite-dimensional input space. The extra conditions in Theorem 6.2.8 that allowed us to relieve the assumption of a finite-dimensional input space are of no help in this chapter: the restriction on the dimension of U is necessary because we will apply the results on the Nehari problem of the previous section, and those are only valid for finite-dimensional input and output spaces. The systems in our class have transfer matrices that have entries in the quotient field HOC [Hoc] ~ 1 - We will denote the class of matrices of any size with components in the field F by MT. For a plant with a transfer matrix in .MHoofHoo]"1, possessing a normalized doubly coprime factorization, we consider the following class of perturbations. Definition 7.2.1 Let G have a left-coprime factorization (M,N). e-admissible leftcoprime factor perturbations ofG are plants with transfer matrices given by
where A = [ A;v -AM ] € MHoo and \\A\\oo < e. We consider the robust stabilization problem for G: find all K € .MHoopHoo]"1 that stabilize all e-admissible perturbations of G? for a given £ > 0. In addition, we are interested in finding the maximal robustness margin £max, i.e., the largest e for which the above problem has a solution. In this problem formulation, as well as in the remainder of this section, stabilization will mean input-output stabilization. The problem of robust stabilization with respect to normalized coprime factor perturbations is now a classic HOC control problem that admits an elegant solution via the solution of a Nehari problem for a normalized left-coprime factorization of the system transfer function. The most general solution was obtained by Georgiou and Smith in [47] for plants possessing a normalized left-coprime factorization. It is known that not all transfer functions possess coprime factorizations, but that the existence of an input-output stabilizing
Section 7.2. Parameterization of Robustly Stabilizing Controllers
97
transfer function is a necessary and sufficient condition for this (see Smith [95]). The disadvantage of the latter solution is that, although very general, it does not yield explicit formulas for the robustly stabilizing controllers. These depend on explicit formulas for a normalized doubly-coprime factorization of the transfer function and these are not available for irrational transfer functions, in general. Special classes for which these are available can be found in Curtain [23], Curtain, Weiss and Weiss [34] and Curtain and Zwart [37]. Furthermore, we obtained them for a class of strongly stabilizable and detectable systems in Chapter 6. In McFarlane and Glover [72], known formulas for the normalized coprime factorizations were exploited to obtain explicit formulas for the robustly stabilizing controllers in terms of the control and filter Riccati equations of the original system. This elegant finitedimensional result was extended to infinite-dimensional systems with bounded inputs and outputs in Section 9.4 of Curtain and Zwart [36] and to systems with unbounded inputs and outputs in Curtain [23] and Curtain and Pritchard [31]. All of these results assume that the state-space realization is exponentially stabilizable and detectable. A different frequencydomain approach was used by Dym, Georgiou and Smith [43] to solve the problem for a special class of delay systems (see [43]). Our approach is to adapt the theory of Chapter 9 of Curtain and Zwart [36] to our situation, where we replace exponential stability (stabilizability) with strong stability (stabilizability) and the Callier-Desoer algebra of transfer functions A- with the algebra HQQ. First, we translate the robust stabilization problem into an HOC-optimization problem, as in Section 4.1 of McFarlane and Glover [72] and Corollary 9.2.9 of Curtain and Zwart [36]. In a later stage, this optimization problem will be related to a Nehari extension problem. Theorem 7.2.2 Suppose that G e .MHoofHoo]"1 has a normalized doubly coprime factorization over H^. Let 0 < e < 1 and K <E MHoo[Hoc]"1. If K stabilizes G and
then it stabilizes all e-admissible left-coprime factor perturbations of G. Proof Let us introduce the shorthand notation
Using G = M~1N and K = UV'1, we have
By Lemma 6.1.4, F e Hoo(£(y, U x Y)) and by assumption \\F («, < e"1. Now,
98
Chapter?. Robust Stabilization
Because K stabilizes G, by Lemma 6.1.4, (MV - NU)~l € H^. We have that A e HOO, F 6 HOO and || AF|| < 1, so / — AF is invertible over HOQ. Therefore, all the factors in the last expression for (/ - G&K] have inverses in H^. We conclude tha Next, we prove that the closed-loop transfer function is in HOO, using the expression for / — G&K, above:
Furthermore, From this, we conclude that the closed-loop transfer function given by
isin To obtain a parameterization of all robustly stabilizing controllers, we would have to obtain equivalence in the above theorem. To do so it is necessary to construct a destabilizing perturbation. This is possible only by assuming extra smoothness conditions on the behavior of G(s) on the imaginary axis (see the proof of Theorem 1 in Georgiou and Smith [48]). Because our systems do not, in general, satisfy these assumptions, their result is not relevant to our problem. As a consequence, there may be more robustly stabilizing controllers than those that are given by our parameterization. Next, we relate the optimization problem from Theorem 7.2.2 to a Nehari extension problem. The parameterization of all solutions to the Nehari problem in Theorem 7.1.2 then will allow us to parameterize the solutions to the robust stabilization problem. Theorem 7.2.3 Suppose that G € AiHoofHoo]"1 has a normalized doubly coprime factorization over HQQ. Let 0 < e < 1 and K € AlHoopHoo]"1. The following three statements are equivalent. 1. K stabilizes G and
2. K has a right-coprime factorization K = UV
l
over HOO satisfying
3. There exists U, V € A4 HOQ such that V is invertible on some right half-plane and
Section 7.2. Parameterization of Robustly Stabilizing Controllers
99
Proof As in the proof of the previous theorem, we use the shorthand notation
(1 => 2). Let K = U\Vl 1 be a stabilizing compensator for G, then by Lemma 6.1.4, MVi — NUi is invertible over HQQ. Consequently,
is a right-coprime factorization of K if U\, V\ is. Because M, TV, M, N form a normalized doubly coprime factorization of G,
is co-inner, i.e., W(jw)W(ju))* = I for all u; G R. As WQcj) is a square matrix, it is also inner: W(ju)*W(ju>) = I for all u e R. So, substituting X = C/iVf 1 and G = M"1^ in the formula for F, we obtain
Consequently,
(2 =>• 1). Let /if = C7V is satisfied. We define
x
be a right-coprime factorization of K and assume that (7.31)
100
Chapter 7. Robust Stabilization
By (7.31), using the inner property of W, we have
This implies that || £21|L < l-£ 2 < 1. Clearly,^ eH 00 (£( U)) and so/- E2 is invertible over HOC. By Lemma 9.4.2 in [36], \\Ei(I - E2)-l\\200 < (1 -£ 2 )/£ 2 . It can easily be checked that -Ei(I - E2)~l = (M*U + N*V)(MV - NU)-1, which in combination with the identity
which was derived in the previous part of the proof, leads to
To show that K stabilizes G, we have to prove that MV — NU is invertible over H^. This is true because MV — NU = E2 — /, which is invertible over HQO, as was explained above. (2 => 3). This part of the proof is trivial and therefore omitted. (3 => 2). Assume that U, V satisfy (7.32) and that V is invertible on some right halfplane. Now,
So,
Consequently,
which implies that
is a Bezout equation and U and V are right coprime. Because V is invertible on some right half-plane, K := UV~l e MH^H^}-1. D The theorem above reduces the problem of robust stabilization with respect to leftcoprime factor perturbations to the solution of the suboptimal Nehari problem for G = [ —N M ], as stated in (7.32). The solution to the Nehari problem is well known (see Power [87]):
where HQ is the Hankel operator with symbol G (see Section 7.1 for the definition of a Hankel operator). The difficulty in solving this particular Nehari problem is that, unlike the case of exponential stability, the Hankel operator is not compact. In Section 7.1, a parameterization of all solutions to the suboptimal Nehari problem with noncompact Hankel operator (and assuming finite-dimensional input and output spaces) was given.
Section 7.2. Parameterization of Robustly Stabilizing Controllers
101
We want to solve the robust stabilization problem for systems H(A,B, C, D) with finitedimensional inputs and outputs under assumptions (i)-(vi). In this case, G = [ —Np Mp ], where Mp, Np are as defined in (6.20)-(6.21). Let us recall the relevant formulas,
where R = I + DD*, LP = -(PC* + BD*)R-\ ALp = A + LPC and P is the unique, self-adjoint, strongly stabilizing solution of the Riccati equation
We need some technical results for the controllability and observability Gramians LB and LCLemma 7.2.4 Let G have a normalized left-coprime factorization G = M~1N. The Hankel operator with symbol G — [ —N M ] satisfies \\Hg\\ < 1. Proof Let us denote U = UxY and F = Y (i.e., G(s) :U-+Y). Clearly, \\H^\\ = \\Hg~ || < || A£~ || = 1, where the first equality follows from the fact that H^, = Hg~ and the last equality follows from the fact that for an inner function F, || AF/|| = |/|| • Suppose now that ||^~|| = 1. Then there must exist {/n}neN with fn e H 2 (y), ||/n|| = 1 such that lirrin^oo \\H^fn\\ - 1. We define gn, g+, g~ by
where TT is the orthogonal projection from L 2 (— joo,joo;U) onto H2(U). Clearly, g+ — Hg~fn and ||^+1|^2 + 11^11^ = \\9n\\l2 = 1, where the last equality follows from the inner property of G~. Thus, we have lirrin-yoo ||svJ"||H2 = 1 andlim n _ >00 ||^~ ||H_L =0. For s — —jcj, we have
Multiplying by [ — Y(ju)* X(jut)* ], where X and Y are the Bezout factors related to the coprime factorization, we obtain
The left-hand side has an extension to a function in H2(Y), the first term on the right-hand side has an extension to a function in H^Y) and the second term on the right-hand side converges to zero. This contradiction implies that H^^H ^ I.
102
Chapter 7. Robust Stabilization
Lemma 7.2.5 Consider the system E(v4, jB, C, D) with finite-dimensional input space and output space, under assumptions (i)-(vi). Let G(s) = [ —Np(s) Mp(s) ] = D + C(sl — A)~1B. The controllability and observability Gramians ofG(s), LB and LC, are given by
where Q and P are the strongly stabilizing solutions of the control and filter Riccati equations (6.14), (6.15). Moreover,
Proof From Lemma 7.2.4, we obtain that r(LsLc) = r(LcLjg} = \\Hg\\ < 1. Hence, I — LsLc and / — LC^B are boundedly invertible. The proof of Lemma 9.4.10 in [36] applies to prove that / + PQ is boundedly invertible and that (7.38), (7.39) hold. The equality of the spectral radii in (7.40) is then trivial. D Because P is a strongly stabilizing solution to the filter Riccati equation (7.36), the system H(A*,C*^B*,D*) is a strongly stable system and, consequently, the dual system E(^4, .B, C, D) is input stable, output stable and input-output stable. Thus, the assumptions H1-H4 in Section 7.1 are satisfied, and we can apply the results of that section to find a parameterization of all solutions to the suboptimal Nehari extension problem for GQ(S):= C(sl — A)~1B. We will formulate the results of Theorems 7.1.1 and 7.1.2 for this special case. Theorem 7.2.6 Consider the system E(yl,5,C, D) with finite-dimensional inputs and outputs under assumptions (i)-(vi). Let A,B,C,D be given by (7.37) and define GQ(S) = C(sl - A}~1B = [ -Np(s) Mp(s) ] — D. The following three statements are equivalent. 1. There exists a J(—s) € M.HOO such that
2. r? (LB LC ), where LB and LC are the controllability Gramian and observability Gramian of£(A,B, C), respectively. 3. There exists an X satisfying the factorization
almost everywhere on the imaginary axis, which is such that Xn (—s) G AiHoo and
Section 7.2. Parameterization of Robustly Stabilizing Controllers
103
Moreover, all J(—s) G -MHoo satisfying (7.41) for a > r? (LsLc] are given by J(-s) = Ri(-s)R2(-s)-1 with
for some Q(—s) G .MHoo satisfying \\Q\\oo < 1. An X satisfying (7.42) and its inverse are given by
where Now we can proceed as in Chapter 9 of [36] to obtain a parameterization of stabilizing controllers for G(s). Theorem 7.2.7 Consider the robust stabilization problemfor ^(A,B,C,D) with finitedimensional inputs and outputs under assumptions (i)-(vi). u
1. The maximum robustness margin is 2. All controllers
given by
for arbitrary L G HQQ satisfying ||L||oo < 1 ore robustly stabilizing for G(s) — D + C(sl — A)~1B, achieving a robustness margin 0 < e < £max, where
and P and Q are the unique, self-adjoint, strongly stabilizing solutions of the control and filter Riccati equations (6.14), (6.15), respectively. Proof The proof is a matter of manipulating the formulas and is a straightforward extension of the proofs of Lemmas 9.4.13 and 9.4.14 and Theorem 9.4.15 in [36], A popular choice in the literature is to use the central controller, which amounts to the choice L(s) = 0 in (7.45). Corollary 7.2.8 Consider the system E,(A,B,C,D) under the assumptions of Theorem 7.2.7. Let P, Q, FQ and W be as in the previous theorem. The controller
104 where
Chapter?. Robust Stabilization and
stabilizes the system with a robustness margin of£,Q<£< £ma,xFinally, we take a look at the important special case of systems in the S(^4, B, B*} class satisfying A* = —A and D = 0. The examples in Sections 9.1 and 9.2 have this special form, and the example in Section 9.3 has this form if its parameters satisfy do = d\ = 0. Corollary 7.2.9 Consider a system S (A, B, B *) satisfying A1-A5 of Section 2.2 and with the additional assumptions that U is finite-dimensional and A + A* = 0. The robustness margin satisfies
and all controllers given by
are robustly stabilizing controllers for G(s) = B*(sl — A) 0 < e < £max, where
1
B, with robustness margin
In particular, the controller K = —I, which corresponds to L(s] — e/\A — £2I, is robustly stabilizing with the maximal robustness margin of l/\/2- Furthermore, the central controller is given by
where Proof It is easy to see that with D = 0 and A + A* = 0, the solutions of the Riccati equations must satisfy P = Q — I. All the results then follow directly from Theorem 7.2.7. Ihat K = — I stabilizes with the maximal robustness margin was also proven in Curtain and van Keulen [33]. The most important class of systems satisfying conditions (i)-(vi) is that of strongly stabilizable and detectable systems T,(A,B,C,D), where A has compact resolvent. For these systems, by Theorem 6.3.3, the robustly stabilizing controllers K(s) in Theorem 7.2.7 yield a strongly stable closed-loop system if K(s] has a strongly stabilizable and strongly detectable realization. In particular, the central controller KQ(S) for systems £(vl,£,.£?*) with A + A* = 0 has a statically stabilizable (even strongly stable) realization £(A — from Corollary 7.2.9. Therefore, this controller will strongly stabilize the systems E(A,B,B*) with A + A* = 0.
Chapter 8
Robustness with Respect to Nonlinear Perturbations A common way to stabilize a linear system is by the use of linear feedback, but in implementing this linear feedback, nonlinear effects may be introduced. The question then arises whether these nonlinearities could render the closed-loop system unstable. In this chapter, we examine this question for strongly stabilizable infinite-dimensional systems. In Chapter 9, we present examples of models of engineering systems that are not exponentially stabilizable, but merely strongly stabilizable. These examples show that there is a need to extend results concerning robustness of asymptotic stability in the presence of nonlinear perturbations to the case of strongly stabilizable systems. This is especially the case because many of the actuators and sensors in structural control are based on so-called smart materials, for instance, piezoelectric and magnetostrictive materials, which show strong nonlinear effects such as saturations, hysteresis and deadzones. If a system is strongly stable but not exponentially stable, then it will have poles arbitrarily close to the imaginary axis. In this sense, there is no stability margin. This may lead one to think that strong stability is a very nonrobust property, but this is not the case. In Chapter 7, we have shown that robustness with respect to linear coprime factor perturbations can be obtained. In this chapter, we prove robustness with respect to certain classes of static and dynamic nonlinear perturbations. A classical mathematical formulation of the problem of robustness of asymptotic stability in the presence of nonlinear perturbations is the absolute stability problem: Under what conditions on the stable system E will the feedback connection as in Figure 8.1 of the system with a nonlinearity remain stable for all functions > in a given class of nonlinearities? We present one famous criterion of absolute stability. We prove a version of the celebrated Popov criterion for a class of strongly stable systems in Section 8.2. As an application of this criterion, we derive a result about asymptotic tracking of constant reference signals using integral control. However, before presenting these robustness results, we introduce in Section 8.1 the necessary concepts related to nonlinear differential equations on Hilbert spaces. Another interesting result in the literature that can be interpreted as a type of robustness of asymptotic stability with respect to nonlinear perturbations is from Slemrod [94], who showed that the feedback u = — y strongly stabilizes a system in the Y^(A,B,B*) class, even in the presence of saturation effects. This very nice result, which is true only for a
105
106
Chapters. Nonlinear Perturbations
Figure 8.1: Feedback interconnection of linear system £ and nonlinearity <j). scalar control, is not included in our results in this chapter, nor does it include our results: they complement one another.
8.1 Preliminaries We give some definitions concerning various solution concepts and stability concepts for a nonlinear differential equation on a Hilbert space Z. We use the following solution concepts, which have been taken from Pazy [85, Chapter 4, Definitions 2.3 and 2.8]. Definition 8.1.1 Let A generate a Co-semigroup T(t] on Z. A function z(t) € C(0,ti; Z) is a mild solution of the differential equation
on [0,ti] if it satisfies the integral equation
If a mild solution is differentiate almost everywhere, z(t) € Li(0,£i; Z), and it satisfies (8.1) almost everywhere in (0,ti), then it is called a strong solution. There are many different possible notions of stability for nonlinear differential equations on Hilbert spaces. Here, we define the types of stability in which we will be interested. The terminology was taken from the book by Willems [119]. The precise formulation for the infinite-dimensional case is as in Wexler [115, 116], who studied the Popov criterion for a class of exponentially stable infinite-dimensional systems occurring as models of a nuclear reactor. Definition 8.1.2 Consider the nonlinear differential equation (8.1). Assume that it has the origin as an equilibrium point, i.e., /(0,£) = 0 for all t > 0. If (i) for any ZQ 6 Z, (8.1) has a unique mild solution z(t) defined for t € [0, oo), and (ii) there exists a strictly increasing continuous function tr : R+ —>• R+ with 7r(0) = 0 such that for any solution z(t] to (8.1) with initial condition ZQ and any r > 0,
then the system is called stable in the large. If, in addition, (Hi) for any ZQ e Z the unique mild solution z(t) —>• 0 as t —> oo,
Section 8.2. The Popov Criterion
107
then we call the system globally asymptotically stable. The system is called uniformly asymptotically stable in the large if the conditions (i), (ii) and (Hi) hold and also (iv) for any ZQ in a bounded set B € Z, each solution z(t) —» 0 as £ —>• oo uniformly with respect to ZQ € B.
8.2 The Popov Criterion In this section, we prove a version of the Popov criterion for strongly stable infinitedimensional systems. This criterion is concerned with the asymptotic stability of a stable linear system connected in feedback with an integrator and a nonlinear function >. It states that a sufficient condition for the closed-loop system to remain asymptotically stable for all 4> in the sector [0, oo) is that the transfer function of the linear system is strictly positive real. This criterion was derived by Popov, and his article [86] was the starting point for many other papers. Narendra and Taylor devoted a large part of their monograph [79] to it. A short and very accessible explanation of the Popov criterion can be found in Vidyasagar [105]. In Wexler [115,116] and Bucci [20], extensions of the Popov criterion to various classes of infinite-dimensional systems were developed, but all these results had the crucial assumption that the linear system is exponentially stable. The nonlinearities under consideration in this problem are so-called sectorial nonlinearities, i.e., their graphs are contained in a cone. A function R is said to be in the sector [a,0] if ax2 < x 0. In that case, its graph is contained in the first and third quadrants. This class of nonlinearities includes two of the most important nonlinearities occurring in practice: saturations and deadzones. To illustrate the nonlinearities for which stability is obtained, we give two graphs of typical nonlinearities. The function in the left-hand graph in Figure 8.2 satisfies y0(y) > 0. For this kind of nonlinearity, we obtain stability in the large and convergence of z(t) to zero. The right-hand one is a graph of a function satisfying y4>(y) > 0 for y 7^ 0. It is for this type of nonlinearity that we obtain uniform asymptotic stability in the large. Next, we formulate our version of the Popov criterion. Theorem 8.2.1 Consider the strongly stable, single-input, single-output system
Figure 8.2: Graphs of typical nonlinearities.
108
ChapterS. Nonlinear Perturbations
with transfer function G(s], in feedback connection with the nonlinearity
Let (f>: R —)• R be any function satisfying the following conditions:
for some strictly increasing continuous function l(r). Furthermore, assume that G(s) is strictly positive real. Then, the system is stable in the large and z(t) -> 0 as t —> oo. If, in addition, y4>(y) > 0/or y ^ 0, then the system is uniformly asymptotically stable in the large. The proof of this result is given in Section 8.2.1. It uses the fact that if S(A,B, C,d) is strictly positive real, then d > 0 and there exists an e e (0, d) such that H(A, B, C, s) is also strictly positive real. From the strict positive real lemma (see Section 4.5.2), it follows that there exists a strongly stabilizing solution P — P* > 0 to the following Riccati equation forz£V(A):
The approach is similar to that in Wexler [116] and Bucci [20]. However, in those papers the exponential stability property of the semigroup is used in a crucial manner. Since we do not have this property, we need to modify the proofs at several steps. Let us comment on the class of nonlinearities. By (8.4), the nonlinearities are in the sector [0,oo), i.e., they have graphs lying in the first and third quadrants. By continuity, this implies that >(0) = 0. Equation (8.6) is a local Lipschitz condition. This condition is necessary to ensure the existence of a strong solution. As mentioned in the introduction, this class of nonlinearities does include saturation effects and deadzones. For the special case of systems of the form %(AB,B, B*,d) under assumptions A1-A5 of Section 2.2 with d > 0 and where AB = A — BB*, the strong stability and strict positive real property are automatically satisfied by properties P1-P6 of Lemma 2.2.6. Therefore, for these systems we obtain the following corollary. Corollary 8.2.2 Consider the system E(^4jg,S,5*,d) underassumptions A1-A5with U = R and d > 0 in connection with a nonlinearity (j> as in (8.3). If $ : R —> R satisfies (8.4)-{8.6), then the system is stable in the large and z ( t ) —> 0 as t -> oo. Under the additional assumption that y(y) > 0 for y ^ 0, the system is uniformly asymptotically stable in the large. In Section 8.2.2, we apply the Popov criterion to show that integral control can be used to track constant reference signals for strongly stable distributed parameter systems in the presence of sec tori al nonlinearities. Balakrishnan [6] has derived a Popov-like criterion for global asymptotic stability of a flexible beam in the presence of sectorial nonlinearities in the feedback loop.
8.2.1
Proof of the Popov Criterion
We prove this result in a sequence of lemmas, following the general approach in Bucci [20].
Section 8.2. The Popov Criterion
109
Lemma 8.2.3 The system (8.2)-(8.3) has a unique strong solution ( z ( t ) , a ( t ) ) on the interval [0,tmax(z0,a0)) for some t m a x ( z 0 , f f 0 ) < oo. Iftmax(zQ,o-0) < oo, then
Moreover, a € Cl([Q,tmax(z0,a0))}Proof We formulate the extended system
and note that this system satisfies the assumptions of Theorem 1.4 of Pazy [85, p. 185]. This theorem shows that (8.8) has aunique mild solution on [Q,tmax(zo,ffo)) and, furthermore, that
By the continuity of a and z and the Lipschitz continuity of 4>, it follows from the expression
that a € Cl([0,T}) for any T < tmax(z0,a0). Hence, 0(
and so
Since ( z ( t ) , a ( t ) } is a mild solution, it is continuous. Consequently, 4>(ff(-)) is continuous as well. Hence, a e C1([Q,tmax(zQ,cr0))). Lemma 8.2.4 //||(z 0 ,0o)|| < r, then
Proof Let
where P is the nonnegative solution of (8.7). Note that V(0,0) = 0 and V(z,a) > 0. Consider (8.8) for initial conditions z(0) = 2oi cr(0) =
110
Chapters. Nonlinear Perturbations
along the trajectories of (8.8), assuming first that ZQ € T>(A). Then, for almost all t € [Q,tmax(zQ,<7o)), there holds
Note that the above result is nonpositive because we took e < d. So, for ZQ € 'D(A) and almost all t e [0,£ maz (zo»0'o))> we have
and integrating from 0 to t gives
This extends to all t e [0,£ max (2:o,cro)) and ZQ e Z. Note that (8.4) implies that both for a < 0 and a > 0, we have
where we have used 0(0) = 0 and (8.6). Similarly, for r > a > 0,
Thus, (8.11) implies that for 0 < t < tmax(zQ,crQ) and ||(2o,cr0)|| ^ r >
This implies (8.9). Lemma 8.2.5 The system (8.2)-(8.3) has a unique strong solution on [0, oo). Furthermore, there exists a strictly increasing, continuous function TT(-) : R + —» R + w///i 7r(0) = 0 such that ||(2(t),cr(t))|| < 7r(r) if\\(z0,a0)\\ < r.
Section 8.2. The Popov Criterion
111
Proof Suppose that IK-Zojtfo)!!
Then, g is nonincreasing on R , 5 is nondecreasing on R + , g is continuous and
Note that from (8.12) it follows that for || (a0, ZQ) || < r,
Combining (8.12) and (8.13) with the fact that g is continuous on R, nonincreasing on R~ and nondecreasing on R + , we can conclude that there must exist a continuous, nondecreasing function 77: R + -> R + such that for a e R,
Hence, if || (z0, ^o) II < r, then for all t > 0,
Next, we want to find a similar estimate for \\z(t) \\. First, we derive an estimate for the L,2-norm of <j)(a(-}}. From (8.11), we have
where we have used (8.12). Now,
where M = supt€[0)00) ||T(t) || is finite by the strong stability of T(i) and l> is the extended input map of S(A, B, C), which is bounded by the strong stability of this system.
772
ChapterS. Nonlinear Perturbations Combining the bounds for |cr(t)| and ||z(£)ll» we obtain fort € [0,tTnax(zo,<7Q)),
where TT : R + —> E+ is a continuous, strictly increasing function with 7r(0) = 0. It now follows from Lemma 8.2.3 that tmax (z0, (a(-)). In the proof of Lemma 8.2.5 it was shown that 0(a(-)) € L<2(0,oo). From (8.8), we have
The fact that >(cr(-)) 6 L 2 (0, oo; U) together with the strong stability of T(t) and the input stability of E(A,£?,C) shows that z(t] -> 0 as t -> oo by Lemma 2.1.3. Next, we show that h(t) — |0(cr(t))| 2 is uniformly continuous on R + . Recall from the proof of Lemma 8.2.5 that if (K^o^o)!! < r > men \cr(t)\ < r)(r). Thus,
where the two last inequalities hold by (8.14) and (8.6). In Lemma 8.2.3, we showed that a is in Cl([Q,tmax(z0,cro))} and
Now, from (8.9) we obtain
by (8.6), (8.14) and (8.15). So, h(t) is uniformly continuous and with (8.15) we can apply Barbalat's lemma (see Corduneanu [22]) to obtain
which implies that a tends to 0-1(0). In the case that y(y) > 0 for y ^ 0, 0~ 1 (0) = {0}, which implies that a(i) -> 0 as t —> oo. Property (iv) can be shown as in Wexler [115, Theorem 1, Step II].
8.2.2
Asymptotic Tracking
In this section, we apply the Popov criterion (Theorem 8.2.1) to prove a result about asymptotic tracking of constant reference signals by integral control, in the presence of monotonic nonlinearities in the loop. There is an enormous amount of literature about tracking by integral control and robustness of tracking with respect to nonlinear perturbations. For instance,
Section 8.2. The Popov Criterion
113
in the infinite-dimensional case, Logemann, Ryan and Townley [70] showed that if G(s) is the transfer function of a single-input, single-output, exponentially stable linear system and (7(0) > 0, then using low gain integral control means that the output y(t) converges to the constant reference signal r as t -> oo for all nondecreasing locally Lipschitz nonlinearities 0, provided that satisfies a certain compatibility condition. The setting of the problem can be represented schematically as in Figure 8.3.
Figure 8.3: Configuration for tracking. We prove an asymptotic tracking result like that in [70] for strongly stable systems S(A, B, C, D) with strictly positive real transfer function G(s). The nonlinearities 0 : R —>• R that we allow are nondecreasing and locally Lipschitz; i.e., there exists an /(r) > 0 such that for all <7i,<7 2 6 R, with absolute value less than r,
These nonlinearities include very important perturbations such as saturations anddeadzones. Theorem 8.2.6 Consider the block diagram of Figure 8.3 under the following assumptions: 1. H(A,B,C,D) is strongly stable and its transfer function G(s] is strictly positive real; 2. 0 : R —>• R is locally Lipschitz and nondecreasing; 3. A is invertible and r is constant, so >r := G(0)-1r is well defined; 4. there exists a a € R such that 0(cr) = • oo. Proof The system is given by
We transform the problem in two steps into the problem of the Popov criterion. First, we transform
114
Chapters. Nonlinear Perturbations
Now, the system reads
Condition 4 of the theorem ensures the existence of a number a such that ij}\(&) — t/v := —(j) r = —G(0)~1r, so we may introduce the variables
Moreover, we introduce the new function i(> given by
Thus, we transformed the tracking problem into the problem of asymptotic stability of
The condition that (f> is locally Lipschitz and nondecreasing implies that tjj is also locally Lipschitz and nondecreasing and, hence, ip satisfies a condition like (8.6). By construction, V'(O) = 0 and y^(y) > 0 for all y 6 K. We can now apply Theorem 8.2.1 to our transformed delta system to show that for t tending to infinity, tp(cr^(t)) —» 0 and z^(t) —> 0, which implies that £/A(<) —» 0 and so y(i) —> r as t —> oo. If A generates a strongly stable semigroup, then on the imaginary axis A has neither eigenvalues nor a residual spectrum (see Arendt and Batty [2]). If, in addition, A has compact resolvent (so its spectrum consists only of eigenvalues), then the imaginary axis is contained in the resolvent set. Thus, in this case, the invertibility condition for A in the above theorem is satisfied. Of course, our result is somewhat different from the one in Logemann, Ryan and Townley [70]. They did not assume that the system was strictly positive real, which forced them to use low gain control: the possible gains for the integral controller are related to those k > 0 for which 1 + kHe[G(s)/s] is positive real. Our assumption that G(s) is strictly positive real removes the necessity of using low gain control.
Chapter 9
Examples of Strongly Stabilizable Systems In this chapter, we present four examples of models of simple engineering systems. All our examples are described by a system of partial differential equations and ordinary differential equations, and three of them have a state-space representation £,(A,B,B*, D) on a Hilbert space Z, satisfying assumptions A1-A5 in Section 2.2. These examples show that models of partial differential equations from various physical domains and with various spatial dimensions can be cast into the T,(A,B,B*) framework. The fourth example is a hybrid system consisting of the parallel connection of a E(A,S,5*) system with a finite-dimensional system. The first example, which was introduced in Example 2.2.4, describes the transverse vibrations of a flexible beam, where control takes place through a piezoelectric film. The second example is a partial differential equation describing the vibrations of a two-dimensional object; it is a model of boundary control of a vibrating rectangular plate. The third example describes the control of sound in a one-dimensional wave-guide, where the ends react as linear oscillators to the acoustic pressure. We have worked out the approximation procedure for solutions of the standard LQ Riccati equation described in Chapter 5 for this example. The fourth and last example is a general model for a flexible structure with rigid body dynamics. Note that even though the first three models have boundary control and boundary observation, they are still described by state-space models with a bounded control operator B and a bounded observation operator C (which will turn out to be equal to B*).
9.1 Boundary Control of a Flexible Beam This section studies Example 2.2.4 in more detail. The model of this example describes the transverse vibrations of a beam of length L, which is clamped at one end and has a point mass attached at the tip. A piezoelectric film is bonded to the beam, which applies a bending moment to the beam if a voltage is applied to it. This voltage is the control input of the system and the angular velocity at the tip is measured as output. This system was described by Bailey and Hubbard [3] as a model of one of the arms of a satellite, consisting of a central hub with four flexible beams attached to it. In their paper,
115
776
Chapter 9. Examples of Strongly Stabilizable Systems
three different controllers were designed as active dampers for this system and tested in numerical simulations and on an experimental test structure. The model they used is given by the partial differential equation
with boundary conditions
and measurement
In Slemrod [94], a state-space representation of this model was provided and analyzed. To obtain such a first-order representation, a number of spaces is needed. The input space and output space both equal R and are denoted by U. Let z = col(zi, 22,23,24). The state space is the Hilbert space
with the inner product
We define the operators A : T>(A] C Z ->• Z and B <E C(U, Z) by
With these definitions of the Hilbert space Z and the operators A and B, the system (9.1)-(9.5) can be represented by the abstract differential equation on Z,
Section 9.2. Boundary Control of a Flexible Plate
117
Slemrod [94] has shown that A is a closed, densely defined, skew-adjoint operator with compact resolvent (in fact, A is invertible and A~l is compact). Furthermore, A generates a Co-semigroup of contractions T(t) on Z. Slemrod computed the eigenvalues of A and showed that there exists an orthonormal basis in Z of eigenvectors of A. The eigenvalues of A are purely imaginary and of the form A = ±j/J?, where n satisfies the equation
Hence, it can be concluded that A is a Riesz-spectral operator (for details on Riesz-spectral operators, see Section 2.3 in Curtain and Zwart [36]). Slemrod showed that £(A —,B*) is approximately observable for appropriate (generic) choices of the length of the beam L. Because A is a Riesz-spectral operator, we can use Theorem 4.2.3 in Curtain and Zwart to show that S(A, —,B*} is approximately observable if and only if T,(A*,B, -) is approximately controllable. In our case, A* = -A and E(A,—,B*) is approximately observable, so we have that E ( — A , B , — ) is approximately controllable. It is an easy application of Theorem 4.2.3 in Curtain and Zwart that S(A,S, —) is then approximately controllable, for generic choices of L, as well. Summarizing, the model (9.1)-(9.5) can be represented by T1(A,B,B*) as in (9.6). Furthermore, £(A B, B*) satisfies conditions A1-A5 in Section 2.2 and, as a consequence, it satisfies properties P1-P5. In particular, this means that this system is strongly stabilizable while it cannot be exponentially stabilizable, because B is of finite rank and bounded and A has infinitely many poles on the imaginary axis!
9.2 Boundary Control of a Flexible Plate In this section, we look at the example of a partial differential equation on a two-dimensional spatial domain. The model is taken from You [120]. It describes the transverse vibrations of an elastic square plate with an edging mass along one edge. The model consists of the twodimensional Petrovsky equation for the dynamics of the plate, coupled with the ordinary differential equations describing the dynamics at the edge where the mass is attached. Consider a thin, homogeneous, square elastic plate with one clamped edge F0 and two simply supported edges FI and Fa. The last edge, Fa, is subject to dynamical equations involving the inertial force, inertial torque, external control force f i ( y , t ) and external control torque /2(y,t). The latter two constitute the input u(t) = col(/j(t), /2(*))Let fJ = [0,1] x [0,1] and let w(x,y,t) be the transverse displacement of the plate for ( x , y ) e fi. Define The following is a suitable model for the vibrating plate:
with initial conditions
118
Chapter 9. Examples of Strongly Stabllizable Systems
and boundary conditions
where a is the Poisson ratio. If we measure the linear and angular velocities along F2, we obtain the measurement
To obtain a suitable first-order formulation, we need to introduce a number of spaces and operators. The following Hilbert space will be used to construct the state space:
with inner product
Next, we introduce the Hilbert space Z\ with inner product {, )z1',
Section 9.2. Boundary Control of a Flexible Plate
779
and the operator
Finally, the state space is the Hilbert space Z = Z\ x Z0 and the system operator A: T>( A) -» Z is defined by
where T>(A) = T>(A\) x Z\. The input space is the Hilbert space
with the usual inner product, and the input operator B : U -> Z is defined by
where BO : U —> ZQ is given by
In You [120], it is shown that A : T>(A) C Z -> Z is a closed, densely defined skewadjoint operator with compact resolvent (A~* € C.(Z} is compact). A generates a unitary Co-group of operators T(t) on Z. The system (9.7)-(9.10) can be reformulated as
for z(t) e Z, with u and y e L 2 (0,1) x L 2 (0,1) and B 6 C(U,Z). In the same paper, it is proved that A — BB* generates a strongly stable contraction semigroup TB(£). Since A has compact resolvent, this implies that T^(t} is also strongly stable. You also shows that T,(A,B,—) is approximately controllable by appealing to results in Benchimol [19]. Here, we prove approximate controllability and observability by appealing to properties of Lyapunov equations from Hansen and Weiss [53] (see also our Lemma 2.1.4). It is clear that the following Lyapunov equations have the solution \I\
Since Tg (t) and TB (£) are strongly stable, LC = \I — LB are the unique solutions of (9.14) and (9.15), respectively. From Lemma 2.1.4, we conclude that Y,(A — BB*,B,B*) have identical controllability and observability Gramians LC = \I = Ls- As in Theorem 4.1.22 of Curtain and Zwart [36], this shows that E(^4- BB*,B,B*) is both approximately controllable and observable. Since these properties are invariant under feedback and output injection, respectively, this implies that Y*(A,B,B*) is both approximately controllable and observable. The above analysis shows that this state-space realization satisfies assumptions A1-A5 in Section 2.2.
120
Chapter 9. Examples of Strongly Stabilizable Systems
9.3 Propagation of Sound in a Duct Our third example describes an acoustical system. It concerns the (one-dimensional) propagation of sound in a duct. The ends of the duct act as a linear oscillator to the acoustic pressure inside. Actuating and sensing takes place through these linear oscillators. A system of this kind can be used to model frequency-dependent reflection of an acoustic plane wave at a so-called locally reacting surface (see page 263 of Morse and Ingard [78]). A number of similar models with different acoustical boundary conditions is described in Banks, Propst and Silcox [12]. We assume that the flow is uniform in a cross section of the duct. Hence, the sound propagates only in the longitudinal direction and the quantities depend only on the longitudinal spatial variable x. ^(x,t) denotes the velocity potential, so that d^/dx(x,t) is the particle velocity in the fluid and d^/dt(x,t) is the acoustic pressure. Now, 3> satisfies the wave equation
The displacements of the two ends ?7o(£)
where f i and /2 are external forces at the ends. From continuity of velocity at the boundary, we obtain
The measurements are taken to be proportional to the velocities at the endpoints,
The model could be brought into first-order form straightforwardly by introducing 2 = col(zi, 22,^3,24) as state vector, where 21 = \?, 22 = d^/dt, z3 = col(?7o,^i) and 24 = col(»7o,»7i). The state space is then endowed with the norm corresponding to the energy of the system (see Beale [17]). For the approximation that we want to perform, a different state-space realization introduced by Ito and Propst [56] is more convenient. An important difference between the approach of Ito and Propst and that of Beale is that the former authors obtain an A-operator that does have compact resolvent, but Beale does not. The approach of Ito and Propst is to decompose \t into a part w+ propagating in the positive direction and a part w~ propagating in the negative direction:
Section 9.3. Propagation of Sound in a Duct
121
with boundary conditions
The state space is taken to be ZQ = L2(0,1) x 1^(0,1) x E4 with inner product (writing Z = CO\(w~(x)1W
+
(x),(^o,(j}i,'t/}0,tl}i))
The input and output spaces are U = Y = R 2 . Let us define the operators AQ, BQ and Co as
722
Chapter 9. Examples of Strongly Stabilizable Systems
With these definitions, the system can be written as
It is an easy computation to see that CQ = BQ . In Ito and Propst [56], it was shown that AQ generates a Co-semigroup of contractions TO(£) on ZQ and AQ has compact resolvent. Because A0 has compact resolvent, A0 has only point spectrum and it can be computed that A € cr(Ao) if and only if A satisfies
All the eigenvalues have negative real part and occur in complex conjugate pairs. The associated eigenvectors are given by
The vectors {z\ \ A 6 a(Ao)} form a complete orthogonal set in ZQ, and all eigenvalues are simple. Because, in acoustics, one is interested only in variations of the pressure, we have to add an extra condition to "filter out" states that correspond to the hydrostatic pressure, i.e., the part of pressure that is constant in the spatial variable x. It can be checked that this corresponds to states that are in the subspace spanned by the eigenvector corresponding to A = 0, i.e., multiples of ZQ = co\(kQki,koki,—2pQki,—2piko,0,0). This leads to the extra condition
Therefore, we do not use ZQ as our state space, but the quotient space
Z is a closed, linear subspace of ZQ and thus a Hilbert space with the same inner product. Define A = AQ\Z. A is the generator of a Co-semigroup of contractions T(t) in Z because To(t) maps Z into Z (this follows from the fact that ZQ is an eigenvector of A and Z is the orthogonal complement of ZQ in ZQ). A also inherits the properties that it is a Riesz-spectral operator and that it has compact resolvent from AQ. So, A has only a point spectrum and, furthermore, A e a (A) if and only if A 7^ 0 and A satisfies (9.20). We define B and C as the restrictions of BQ and Co to Z. Using the controllability and observability test in Theorem 4.2.3 of Curtain and Zwart [36], it can be computed that E(^4, B, C) is approximately controllable and approximately observable. We can summarize the previous results by saying that C = B* and the system E(yl, B,B*) satisfies assumptions A1-A5 in Section 2.2.
Section 9.3. Propagation of Sound in a Duct
123
For this example, we perform a numerical approximation of the strongly stabilizing solution of the standard LQ algebraic Riccati equation,
as was described in Chapter 5. To do so, we construct a sequence of approximating finitedimensional systems, and we will show that it satisfies the conditions that guarantee convergence of the approximating solutions of the finite-dimensional LQ Riccati equations to the solution of the infinite-dimensional LQ Riccati equation. Ito and Propst [56] proposed an approximation scheme for Y,(AQ,BQ,CO). We explain this scheme and then make an obvious modification, to obtain an approximation scheme for Let • We define
where 0j^ = >0, ^ = 0i, V^ = V^. V'f = V'l andu;£",iu+, A; = 0,...,N, m = 0,...,N1, are defined as follows. Let Pk(x) be the Legendre polynomial of degree k. The set {Pfc(ar), k E N) forms an orthogonal basis of L 2 (-l,l) and ||Pfc|li 2( _ 1):l) = 2/(2k + 1). Hence, the polynomials Pk(x) = \/2k + IPk(2x — 1) form an orthonormal basis for L,2(0,l). We expand w~(x) and w+(x) with respect to the basis (pk(x), A; € N} and truncate these expressions:
Note the difference in notation between the approximating function wjj (x) and the coefficient w^f. Let iff denote the corresponding embedding that associates with a vector v in R 2jv+5 an element in Z0 defined by
Next, we introduce
where a and 6 are chosen such that WN(X) and w^f_1(x) satisfy the boundary conditions
124
Chapter 9. Examples of Strongly Stabilizable Systems
We will now use WN(X) and Wpf_1(x) to define the approximation Aff of AQ:
The approximations B^ and CQ of BQ and C0 are given by
The approximating state ZN is now obtained by orthogonal projection of z$ on the 2N+4dimensional linear subspace of R 2 ^ +5 of vectors satisfying
Let this orthogonal projection be denoted by xN and the associated embedding from R2;v+4 into R 2N+5 by hN. The approximating system E(AN,BN,CN) is obtained by restricting Ag, B^ and C0N to this 2N+4-dimensional space, i.e., AN = XNA{?hN, BN = xNB^, CN — C Q } I N . The projection TTN and embedding iN for the total approximation are now given as
Now, TTNiN is an orthogonal projection in Z and iNnN = /2JV+4. Note that the approximation AN does not satisfy AN — TTN AiN, as in the scheme of Section 5. However, we do have AN + (AN)* < 0 and CN = (BN)*. So, the result of Lemma 5.4.1 is still valid. In Ito and Propst [56], it was shown that for every z e T)(A0) there exists a sequence
Section 9.3. Propagation of Sound in a Duct
125
and
as N —> oo. This is easily adapted to the existence of a sequence ZN G R 2Ar+4 converging to z € T>(A). Similarly, we can prove that for all z € T>(A*} there exists a sequence ZN e R 2AT+4 such that
and
It is easily shown that CN and BN converge strongly to C and B, respectively. As in Lemma 5.4.2, these results imply that the finite-dimensional systems ^(AN,BN,CN) converge strongly to the infinite-dimensional system £(A,B,B*). Hence, we can conclude that if Z(AN,BN,CN) is observable for all N, then the solution X of the infinitedimensional LQ Riccati equation associated with £(^4,5, B*) can be approximated by the sequence of solutions XN of the matrix LQ Riccati equations associated with the systems E(AN,BN,CN). We implemented the approximation scheme in Matlab. The values for the parameters we used are
We computed the approximating systems for N — 2,4,8,16,32,64. Then, we computed the solutions to the Riccati equations corresponding to these approximations. To illustrate the convergence of the solutions, we compare the solutions in a number of characteristics. First, we compute the matrix norms of the approximating solutions. Second, we restrict X64 to the spaces on which the lower-order approximations are defined and compute the norm of the difference, \\XN - irNi6*X647T64iN \\, for N = 2,4,8,16,32. This norm measures to what extent the lower order approximation matches the behavior of the higher order one on the lower-dimensional space. These values are listed in Table 9.1. Finally, we investigated the closed-loop systems corresponding to the different approximative Riccati feedbacks FXN. We have given plots of the closed-loop poles and we have computed the open-loop and closed-loop step responses. Because the step responses for the different values of N were indistinguishable, we give only the plots for N = 16.
2 4 8 16 32 64
0.9802 0.9802 0.9802 0.9802 0.9802 0.9802
6.31e-ll 1.81e-10 6.25e-10 2.32e-9 1.57e-8 0
Table 9.1: Comparison of norms of the different approximations.
726
Chapter 9. Examples of Strongly Stabilizable Systems
Figure 9.1: Closed-loop poles, N = 2,4,8,16,32,64.
Figure 9.2: Close-up plot of closed-loop poles, N = 16,32,64.
Section 9.3. Propagation of Sound in a Duct
Figure 9.3: Open-loop step response, N = 16.
Figure 9.4: Closed-loop step response, N — 16.
127
725
Chapter 9. Examples of Strongly Stabilizable Systems
Figure 9.5: Open-loop step response, N = 16.
Figure 9.6: Closed-loop step response, N = 16.
Section 9.4. A Flexible Structure with Rigid Body Dynamics
129
From these results, we observe the following. The norms of the approximations are very close to each other and far larger than the norm of the difference between the low order approximations and the restriction of X64 to the lower order spaces. This indicates that the low order approximations succeed very well in capturing the behavior of the infinitedimensional solutions on the finite-dimensional spaces on which they are defined. The plots of the closed-loop poles in Figures 9.1 and 9.2 indicate that the closed-loop system matrices converge as N tends to infinity. As expected, since the original system is not exponentially stabilizable, the poles are converging to the imaginary axis. The plots of the step responses for the different values of TV are indistinguishable. We plotted step responses for TV = 16 in Figures 9.3 and 9.4. In Figures 9.5 and 9.6, we plotted the response of the velocity potential ty(x,t) to a step input at x = 0, for both open-loop and closed-loop systems.
9.4 A Flexible Structure with Rigid Body Dynamics This last example gives a general model describing vibrations in a flexible structure combined with the rigid body dynamics of the structure (i.e., the translation of its center of mass and its rigid body rotation). The model we give is an infinite-dimensional version of the one in Sections 2.1 and 2.2 in Joshi [57]. This model is an example of a statically stabilizable system that is not in the £(-4, B, B*) class for two reasons. First, A does not generate a contraction semigroup, and second, C is not equal to B*. It is therefore a motivating example for those results in this book that deal with strong stability for systems Y^(A,B,C) outside the E(A,B,B*) class. The structure is equipped with a number (my) of force actuators and a number (HIT) of torque actuators. Let £ € R3 denote the position of the center of mass and let the force exerted by the ith actuator be fa € R3. The translation of the center of mass is given by
The attitude of the structure is given by the vector a E R3 describing the rigid body Euler angle with respect to the x-, y- and z-axes. It satisfies
where Tj denotes the torque exerted by the jth torque actuator and r^ € R3 describes the location where the force fi is applied. The elastic motion is described by
Here, q is the infinite sequence of modal amplitudes, which is assumed to be an element of a HilbertspaceQ(forinstance,Q = J 2 (N)). F = col(/i,... ,/ m/ ) andT = col(Ii,... ,T mr ). The infinite matrices $i G £(Q, R 3 ) and $., G C(Q, R 3 ) are called the mode shape matrices and mode slope matrices, respectively. A is the stiffness operator and it is assumed that
130
Chapter 9. Examples of Strongly Stabilizable Systems
A = A* > 0. The damping operator D satisfies D — D* > 0. Both D and A are closed and densely defined. In general, both are unbounded operators. We call this a hybrid system because it consist of a finite-dimensional part (the dynamics of £ and oi) coupled with a distributed part (the elastic motion). The coupling consists of the fact that both parts are affected by the same control inputs /» and Tj. The aim of this section is to show that under certain conditions this system has a statically Stabilizable state-space realization. We define p = col(ra£, Ja, q) € M3 x R3 x Q =: Z\. Then the dynamics are described as
where
u = col(/i,...,/ m/ ,Ti,...,T mr ) € R 3m /+ 3m r =: \] and the matrices Ri, i = 1...,m f , are given by
If we have position sensors at the same locations as the force actuators and attitude sensors at the same locations as the torque actuators, then the observation is given by
On the other hand, the use of rate sensors measuring translational and rotational velocities at those locations leads to the observation
Following Joshi [57], let us suppose that both types of measurements are available and the observation is y = col(t/p, yr). To construct a first-order representation, we take as the state z = co\(z,z) on the state space Z = {z = 001(2:1,^2) € Z\ x Z\} with inner product
This leads to the following representation of the systems (9.22), (9.23), (9.24), (9.27), (9.28),
where
Section 9.4. A Flexible Structure with Rigid Body Dynamics
131
To show that the above equations represent a bounded linear system, there is some work to be done. We need to show that 1.
Ce£(Z,UxU);
2. equation (9.29) defines an inner product on Z. This is true only if 3. A is the generator of a Co-semigroup on Z. Let us examine these three issues: 1. The operator F is a bounded operator from Z\ to U because the operators $it tyj and Ri are bounded. Therefore, it is obvious that C"r = [ 0 F ] is bounded. To establish the boundedness of C, we therefore need only to check that Cp = [ F 0 ] € £(Z, U). We have
So, C is indeed a bounded output operator. 2. Next, we prove that A + FTF > 0. From the relation between A and A in (9.26) and the fact that A is assumed to be a positive, self-adjoint operator, it follows that A -f FTF > 0 if and only if
So, defining F = F
r /e i, we require that ker(F) = {0}, where
Suppose that there exists an x € M6 such that Tx = 0. Then,
and
for i = 1,..., rnj. Consequently, x = 0 and, indeed,
732
Chapter 9. Examples of Strongly Stabilizable Systems
3. Next, we show that A generates a Co-semigroup on Z via a perturbation technique. More precisely, we show that A% := A — [ FT 0 ][ 0 FT ]T generates a Co-semigroup Z. As A is a bounded perturbation of A?., this implies that A generates a semigroup on Z, as well. We need the following assumptions: (a) A + r T r is boundedly invertible; (b) £>(A) C T>(D) (or, equivalently, £>(A) c £>(£>)). The domain of definition of A2 is /D(A2) = V(A] = £>(A) x Z>(As). Clearly, A 2 is densely defined, because A is. First, we show that A% is closed. It is easy to see that under assumptions (a) and (b), the operator V given by
satisfies
From Theorem A.3.46 in Curtain and Zwart [36], we can now conclude that A% is a closed operator. Next, we show that both A2 and its adjoint are dissipative operators. For any z <E T>(A2),
Hence, A% is a dissipative operator. In exactly the same way, it can be shown that A%, which is given by
is dissipative, as well. We can now apply the Lumer-Phillips Theorem (see Section 1.4 in Pazy [85]) to show that A2 generates a Co-semigroup of contractions. As a consequence,
generates a Co-semigroup, as well (see Theorem 3.2.1 in Curtain and Zwart [36] for a proof of the fact that A -f D generates a Co-semigroup on Z if A generates a Co-semigroup on Z and D is a bounded operator on Z). We now state our main result concerning this example. Lemma 9.4.1 Let the system E(^4, B, C) be defined as in (9.30) on the state space Z and assume that the following conditions are satisfied: (i) A = A* > 0, £> = £>*> 0 are closed, densely defined operators with V(D] D P(A); (ii) A has compact resolvent;
Section 9.4, A Flexible Structure with Rigid Body Dynamics
133
(Hi) S(yl,B,C1) is approximately controllable; (iv) Y,(A2,B,Cr) is approximately observable, where (v) A + TTr is boundedly invertible and (A + r r r)~ 1 £> is bounded. Then the system S(j4, B,C) is statically stabilizable. In particular, the static output feedback law u = —Gpyp — Gryr renders the closed-loop system strongly stable for any positive definite matrices Gp and Gr. Proof The conditions (i) and (v) ensure that 12(A,B,C) is a well-defined, bounded, linear system. The proof consists of the following steps. 1. We first restrict to the case that Gp = I and show that the preliminary feedback u = —yp + v leads to the system £(j42i B, C). This system has almost all features of systems in the £(A, B, B*) class. The only difference is that there are more outputs, namely, yp. 2. Then we show, using the Y*(A:B,B*) structure and additional information on the system, that the feedback v = -Gryr leads to a strongly stable, closed-loop system E(Ad,B,C), where Acl=A- BCP-BGrCr. 3. Finally, we show how to include the case of general positive definite matrices Gp. Let us go through these steps. For ease of reference, we write
1. Application of the static output feedback law u = — yp + v to the system Si leads to T^(A2,B,C), i.e., the system E2 with the additional output yp = Cpz. We show that S2 is in the H(A,B,B*) class. Earlier in this section, we have already shown that A^ generates a contraction semigroup. As approximate controllability is invariant under feedback, E2 is approximately controllable by assumption (Hi). Assumption (iv) states that ^2 is approximately observable. That A% has compact resolvent follows from the fact that A has compact resolvent, as follows. We have
The first term in the right-hand side is compact for all s € p(A). For s 6 p(A) n p(A2), the second term is the product of a compact operator with a bounded operator; hence it is compact. As both A and A2 are dissipative, the open right half-plane is contained in p(A) n p(A2), so this intersection is not empty. Consequently, the right-hand side is compact and so A2 has compact resolvent. Thus, E2 satisfies A1-A5 (see page 17) and so it is in the E(A,B,B*) class. 2. As E2 = E(A2,B,B*) belongs to our standard class, it is strongly stabilized by the feedback v = -yT. That this also happens for v = —Gryr can be seen by rewriting this case as the feedback v = —y for the system E(A,BS, SB*), where 5 is the square root of Gr. This system is also in the "£(A,B,B*) class. Thus, we have that £3 = E(A2 - BGrCr,B,Cr) = H(Aci,B,Cr) is strongly stable. That is, we have
134
Chapter 9. Examples of Strongly Stabilizable Systems • output stability: • input-output stability:
However, we are interested in the strong stability of the overall closed-loop system Sc( = £(-Ach B, C). Hence, it remains to show that, in addition to the above, output stability and input-output stability hold for the output yp, as well. The remaining analysis of part 2 of this proof is therefore dedicated to proving • output stability with respect to yp: • input-output stability with respect to yp: To do so, we take a closer look at (sI — Aci) ~1. For convenience of notation, let us write Aci = A + T T r and Dcl - D + TTGrT. Then,
If Aci is not exponentially stable, then (si — Ad)'1 <£ HOC. However, from the output stability of £3, we can conclude that
and from the input-output stability,
We nowfirstprove output stability of Sc;, i.e., we show that Cp(sl - Aci)~1z e H2(£7) for all z e Z. From (9.31), it follows that
The first term on the right-hand side is in H2 for all z\, by (9.33). For the second term, we obtain from the assumption that A";1 and Ajj1 Dc\ are bounded and from (9.32) that, for all Zl 6 Zi,
In a similar way, it can be proved that the last term on the right-hand side is in H 2 (f/) for all z^ € Zi, using (9.32) and the boundedness of A";1. Next, we prove the input-output stability of Scj, i.e., we show that the closed-loop transfer function from u to yp, Cp(sl — Aci)~1B, is in H 00 (£(C/)). We have
and we denote the above transfer function by G(s). If we write
Section 9.4. A Flexible Structure with Rigid Body Dynamics
135
then G(s) = F(s)/s. By (9.34), F(s) e HOO(£(£/)). The function 1/s is also holomorphic in CQ" anditis bounded in Co"\Be(0) for any £>0, where we define jBe(0) = {se CQ" | |s| < e}. Combining these observations, it follows that G(s) is holomorphic in CQ~ and bounded in C0~\.Be(0). Furthermore, G(0) = FA^1rT, which is bounded by assumption (v). Thus, G does not have a pole at s = 0 and we can conclude that G is also bounded on Be(0), which shows that G e H00(£(C/')). 3. The case of general Gp can be dealt with by defining the inner product on the state space alternatively as
The whole analysis above goes through without change in this new inner product. From the positive definiteness of Gp, it can easily be shown that the norm induced by this new inner product is equivalent to the original norm. Hence, the system is also strongly stable with respect to the original state space Z D Assumptions (i) and (ii) are satisfied by most, if not all, systems of interest. Assumption (Hi) and (iv) depend on the actuators and sensors implemented. Note that assumption (iv) concerns A2, not A. The reason is that the pair (A, Cr] can never be observable: the position of the center of mass and the attitude of the structure cannot be reconstructed from the measurements of the rate sensor, because the dynamics do not depend on the position and attitude. However, the coupling introduced in A2 makes the dynamics dependent on them, and so it becomes possible to observe the position and the attitude from the rate sensors. Although for certain structures of practical interest assumption (v) will be satisfied, it still is quite restrictive. We used it in the general setup of the example in this section because it leads to a fairly simple proof. We believe that in more specific examples this assumption can be relaxed.
This page intentionally left blank
Chapter 10
Conclusions In the preceding chapters, we have presented a diversity of system and control theoretic results about strongly stabilizable control systems. Here, we give an overview of the main contributions of the book and we discuss some open problems and different research directions. Strong stability: We have introduced a framework for studying strongly stabilizable control systems. This includes definitions of strong stability, strong stabilizability and strong detectability. An important feature of all those definitions is that they comprise both internal and external stability properties. Directly related to this framework are basic system theoretic results such as the equivalence of internal and external stability of strongly stabilizable and detectable systems and the convergence of the state of a strongly stable system when an L,2-input is applied. Within this general framework, we have introduced a special class of systems, the class £(A,B,B*) of dissipative systems with collocated actuators and sensors. These systems are often used as models of flexible structures and other mechanical systems, and their special structure allows us to give stronger results for systems in this class. Riccati equations: We have given conditions for the existence and uniqueness of a strongly stabilizing solution to very general algebraic Riccati equations. We have shown that a Riccati equation associated with a strongly stable system has a strongly stabilizing solution if and only if the Popov function has a Wiener-Hopf factorization. This result was adapted to include statically stabilizable systems. This factorization-based approach allowed us to give versions of the positive real and bounded real lemmas for strongly stable systems and to prove a result about the existence of J-spectral factorizations. For the LQ Riccati equation, we showed that there exists a unique, strongly stabilizing solution if the system is strongly stabilizable and strongly detectable. Furthermore, we developed a procedure for approximating the strongly stabilizing solution of the LQ Riccati equation by solutions of a sequence of finite-dimensional Riccati equations and we proved the convergence of this approximation procedure. Coprime factorizations and dynamic compensators: We studied input-output stabilization and strong stabilization by dynamic compensators for strongly stabilizable and detectable systems. In connection with this, we studied the key question of state-space formulas for normalized doubly coprime factorizations for strongly stabilizable and detectable systems. It is doubtful that a satisfactory generalization of all the known finite-dimensional results can be obtained for strongly stabilizable and detectable systems. However, under the assumptions of compact resolvent and finite-dimensional input and output spaces, we have
137
138
Chapter 10. Conclusions
obtained state-space formulas for normalized doubly coprime factorizations for strongly stabilizable and detectable systems. The problem of dynamic compensators is more difficult to resolve and we have obtained only satisfactory results for the class of statically stabilizable systems. Robustness: Although it has long been believed that strong stability is a very nonrobust property, we have shown that robustness against very general perturbations can be obtained. We have given a parameterization of robustly stabilizing controllers for strongly stabilizable and detectable systems with compact resolvent and finite-dimensional inputs and outputs in the case of coprime factor perturbations. Furthermore, we have shown that the Popov criterion guarantees global asymptotic stability for a feedback system consisting of a strictly positive real, strongly stable system in the forward loop and a sectorial nonlinear perturbation in the feedback loop. Examples: We have collected a number of examples of physical systems that have a strongly stabilizable and detectable state-space representation. Three of these examples are in our class of dissipative systems with collocated actuators and sensors, and for one of them we have worked out numerically the approximation procedure for the strongly stabilizing solution of the LQ Riccati equation.
10.1
Possibilities for Further Research
Even after the nine previous chapters full of theory, we are still far away from completion of systems and control theory for strongly stabilizable systems. There are still many open problems and possible new research directions. First, there are a number of instances where the results in this book are not yet satisfactory. Coprime factorizations: We have obtained state-space formulas of normalized doubly coprime factorizations only under the additional assumptions that A has compact resolvent and U and Y are finite-dimensional. It remains an open problem whether the usual formulas also constitute a normalized doubly coprime factorization for the general class of strongly stabilizable and detectable systems. It is to be expected that a solution of this problem of coprime factorizations for more general classes of strongly stabilizable and detectable systems Y,(A,B,C, D) will immediately lead to a parameterization of robustly stabilizing controllers for these systems. It is, however, doubtful whether such formulas for normalized doubly coprime factorizations can be found for the general strongly stabilizable/strongly detectable case. Robust stabilization: Another problem in the parameterization of robustly stabilizing controllers is the fact that our parameterization does not include all robustly stabilizing controllers. This deficiency is due to the fact that the usual proofs rely on the construction of a destabilizing perturbation. For this, we need continuity of certain functions on the imaginary axis. As our functions are in HQQ, we do not have this continuity. The same open problem was observed in Georgiou and Smith [47]. There, the authors expressed their belief in a solution of this problem, although they acknowledged that a completely different approach would be required. Next, we discuss a number of possibilities for different research directions. Weak stability: During the development of the current research, an intrinsic problem of working with strong stability showed up at several occasions. All instances had to do with the fact that if A generates the strongly stable semigroup T(t), then it may happen that T*(t) is not strongly stable. As a consequence, strong stabilizability is not dual to strong detectability and whereas the LQ state feedback does strongly stabilize the system,
Section JO.1. Possibilities for Further Research
139
LQ output injection may not result in a strongly stable system. A way to avoid this lack of duality would be to concentrate instead on weak stability. Let us elaborate on this idea for a moment. We define a system to be weakly stable if it is input stable, output stable, inputoutput stable and its semigroup is weakly stable. The definitions of weak stabilizability and weak detectability can be derived, in a similar way, from their strong counterparts. In this case, weak stabilizability is the dual of weak detectability. A crucial next step then would be to show whether the application of an arbitrary Lg-input to a weakly stable system leads to a state z(t) that converges to zero weakly (i.e., a generalization of Lemma 2.1.3). We believe that this result holds. With this result, it would probably be possible to extend the theory on the LQ Riccati equation and obtain a result for the filter Riccati equation that is completely dual to the result for the control Riccati equation. Also, the theory for dynamic compensation would not suffer from the lack of duality. The down side is that weak stability is too weak to be useful. Many of the other results in this book could most likely be extended to weak stability as well. Furthermore, we know from Benchimol [19] that if A generates a weakly stable semigroup T(t) and has compact resolvent, then T(t) is in fact strongly stable. So, for systems whose system operator A has compact resolvent, the results of weak stabilization would even imply strong stability. For certain results, one problem could be that a result analogous to the spectral characterization of strongly stable semigroups by Arendt and Batty [2] is needed. We are not aware of such a characterization of weakly stable semigroups. Unbounded operators: An important extension of the theory in this book would be to include the possibility of unbounded input and output operators B and C. This would open up an important class of applications, for instance, in problems of structural acoustics (see Banks and Smith [13, 14]. We have never touched upon this problem, but research in this direction has been initiated by Curtain and G. Weiss. HOC-control: It would be interesting to see if the state-space approach to H^ control, as it was developed for finite-dimensional systems in Doyle et al. [41] and for PritchardSalamon systems by van Keulen [104], could be extended to strongly stabilizable systems. Research in this direction is carried out by Staffans [100] and Mikkola [73] in the more general context of well-posed linear systems. Approximation issues: The approximation procedure for the strongly stabilizing solution of the Riccati equation was presented only for the LQ Riccati equation. It would be very interesting to extend this to more general Riccati equations, like those occurring in HOQ control. In the context of H^ control, the usual procedure is to approximate the system in Lx sense by a finite-dimensional system and to do an H^ control design for the finite-dimensional plant. If the L^ approximation error is sufficiently small, then the closed-loop system is still guaranteed to be stable and there is a certain guaranteed degree of performance for the closed-loop system. This approach is problematic for strongly stable systems because the noncompact Hankel operator causes problems in L^ approximation. It seems, therefore, necessary to develop a different approach to the study of approximation for robust control.
This page intentionally left blank
Bibliography [1 ] V.M. Adamjan, D.Z. Arov and M.G. Krein. Infinite Hankel block matrices and related extension problems. American Mathematical Society Translations, 111:133-156, 1978. [2] W. Arendt and C.J.K. Batty. Tauberian theorems and stability of one-parameter semigroups. Transactions of the American Mathematical Society, 306:837-841, 1988. [3] T. Bailey and I.E. Hubbard, Jr. Distributed piezoelectric polymer active vibration control of a cantilever beam. AIAA Journal on Guidance, Control and Dynamics, 8:605-611, 1985. [4] A.V. Balakrishnan. Strong stabilizability and the steady state Riccati equation. Applied Mathematics and Optimization, 7:335-345, 1981. [5] A.V. Balakrishnan. A Mathematical Formulation of the SCOLE Control Problem. Technical Report CR 172581, NASA Langley Research Center, Hampton, VA, 1985. [6] A.V. Balakrishnan. Stability enhancement of flexible structures by nonlinear boundary-feedback control. In Boundary Control and Boundary Variations, Lecture Notes in Control and Information Science 100, pages 18-37, Springer-Verlag, New York, 1998. [7] A.V. Balakrishnan. Compensator design for stability enhancement with co-located controllers. IEEE Transactions on Automatic Control, 36:994-1007, 1991. [8] A.V. Balakrishnan. Compensator design for stability enhancement with co-located controllers: Explicit solutions. IEEE Transactions on Automatic Control, 38:505507, 1993. [9] A.V. Balakrishnan. Robust stabilizing compensators for flexible structures with collocated controls. Journal of Applied Mathematics and Optimization, 33:35-60,1996. [10] J.A. Ball and J.W. Helton. A Beurling-Lax theorem for the Lie group l/(ra, n) which contains most classical interpolation theory. Journal of Operator Theory, 9:107-142, 1983. [11] H.T. Banks and J.A. Burns. Hereditary control problems: Numerical methods based on averaging approximations. SI AM Journal on Control and Optimization, 16:169208, 1978.
141
142
Bibliography
[12] H.T. Banks, G. Propst and R.J. Silcox. A comparison of time domain boundary conditions for acoustic waves in wave guides. Quarterly of Applied Mathematics, LIV:249-265, 1996. [13] H.T. Banks and R.C. Smith. Models for control in smart material structures. In Identification and Control in Systems Governed by Partial Differential Equations, H.T. Banks, R.H. Fabiano, and K. Ito, editors, pages 26-44, SIAM, Philadelphia, PA, 1993. [14] H.T. Banks and R.C. Smith. Feedback control of noise in a 2-D nonlinear structural acoustics model. Discrete and Continuous Dynamical Systems, 1:119-149, 1995. [15] C.J.K. Batty. Tauberian theorems for the Laplace-Stieltjes transform. Transactions of the American Mathematical Society, 322:783-804, 1990. [16] C.J.K. Batty and V.Q. Phong. Stability of individual elements under one-parameter semigroups. Transactions of the American Mathematical Society, 322:805-818, 1990. [17] J.T. Beale. Spectral properties of an acoustic boundary condition. Indiana University Mathematics Journal, 26:895-917, 1976. [18] C.D. Benchimol. Feedback stabilizability in Hilbert spaces. Applied Mathematics and Optimization, 4:225-248, 1977. [19] C.D. Benchimol. A note on weak stabilizability of contraction semigroups. SIAM Journal on Control and Optimization, 16:373-379,1978. [20] F. Bucci. Frequency domain stability of nonlinear feedback systems with unbounded input operator. Dynamics of Continuous, Discrete and Impulsive Systems, to appear. [21] F.M. Callier and J. Winkin. LQ-optimal control of infinite-dimensional systems by spectral factorization. Automatica, 28:757-770, 1992. [22] C. Corduneanu. Integral Equations and Stability of Feedback Systems. Academic Press, New York, 1973. [23] R.F. Curtain. Robust stabilizability of normalized coprime factors; the infinitedimensional case. International Journal of Control, 51:1173-1190, 1990. [24] R.F. Curtain. The Salamon-Weiss class of well-posed linear systems: A survey. IMA Journal of Mathematical Control and Information, 14:207-223, 1997. [25] R.F. Curtain and A. Ichikawa. The Nehari problem for infinite-dimensional systems of parabolic type. Integral Equations and Operator Theory, 26:29^5, 1996. [26] R.F. Curtain and J.C. Oostveen. Absolute stability for collocated systems. In Proceedings of the International Conference on Optimization: Techniques and Applications (ICOTA), Perth, July 1998, L. Cacetta, ed., Curtin University of Technology, pp. 366-372. [27] R.F. Curtain and J.C. Oostveen. The Nehari problem for nonexponentially stable systems. Integral Equations and Operator Theory, 31:307-320, 1998.
Bibliography
143
[28] R.F. Curtain and J.C. Oostveen. Necessary and sufficient conditions for strong stability of distributed parameter systems. Systems and Control Letters, 37:11—18, 1999. [29] R.F. Curtain and J.C. Oostveen. Normalized coprime factorizations for strongly stabilizable systems. In Advances in Mathematical Control Theory (proceedings of a workshop in honour of Diederich Hinrichsen on the occasion of his 60th birthday), U. Helmke, editor, 1999. [30] R.F. Curtain and J.C. Oostveen. Robustly stabilizing controllers for strongly stabilizable and detectable infinite-dimensional systems. In Proceedings of the IEEE Conference on Decision and Control, pages 4274-A290. IEEE Press, Piscataway, NJ, 1999. [31] R.F. Curtain and A.J. Pritchard. Robust stabilization of infinite-dimensional systems with respect to coprime-factor perturbations. In Control Theory, Dynamical Systems and Geometry of Dynamics, K. D. Elworthy, W. N. Everitt and E. B. Lee, editors, pages 437-456. Marcel Dekker, New York, 1993. [32] R.F. Curtain and A. Ran. Explicit formulas for Hankel norm approximations of infinite-dimensional systems. Integral Equations and Operator Theory, 13:455^169, 1989. [33] R.F. Curtain and B. van Keulen. Robust control with respect to coprime factors of infinite-dimensional positive real systems. IEEE Transactions on Automatic Control, 37:868-871, 1992. [34] R.F. Curtain, G. Weiss and M. Weiss. Coprime factorizations for regular linear systems. Automatica, 32:1519-1532, 1996. [35] R.F. Curtain and HJ. Zwart. The Nehari problem for the Pritchard-Salamon class of infinite-dimensional linear systems: A direct approach. Integral Equations and Operator Theory, 18:130-153, 1994. [36] R.F. Curtain and H.J. Zwart. An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York, 1995. [37] R.F. Curtain and H.J. Zwart. Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems. Systems and Control Letters, 28:11-22, 1996. [38] E.B. Davies. One-Parameter Semigroups. Academic Press, London, 1980. [39] C.A. Desoer and M. Vidyasagar. Feedback Systems: Input-Output Properties. Academic Press, New York, 1975. [40] A. Devinatz and M. Shinbrot. General Wiener-Hopf operators. Transactions of the American Mathematical Society, 145:467^94, 1969. [41] J.C. Doyle, K. Glover, P.P. Khorgonekar and B.A. Francis. State-space solutions to standard H% and H^ control problems. IEEE Transactions on Automatic Control, 34:831-847, 1989. [42] P.L. Duren. Theory ofHp Spaces. Academic Press, New York, 1970.
144
Bibliography
[43] H. Dym, T.T. Georgiou and M.C. Smith. Explicit formulas for optimally robust controllers for delay systems. IEEE Transactions on Automatic Control, 40:656— 669, 1995. [44] C. Foias and A. Tannenbaum. On the Nehari problem for a certain class of Loofunctions appearing in control theory. Journal of Functional Analysis, 74:146-159, 1987. [45] P.A. Fuhrmann. On the Corona theorem and its application to spectral problems in Hilbert space. Transactions of the American Mathematical Society, \ 32:493-520, 1968. [46] P.A. Fuhrmann. Linear Systems and Operators in Hilbert Space. McGraw-Hill, New York, 1981. [47] T.T. Georgiou and M.C. Smith. Optimal robustness in the gap metric. IEEE Transactions on Automatic Control, 35:673-685, 1990. [48] T.T. Georgiou and M.C. Smith. Robust stabilization in the gap metric: Controller design for distributed plants. IEEE Transactions on Automatic Control, 37:1133— 1143, 1992. [49] J.S. Gibson. Linear-quadratic optimal control of hereditary differential systems: Infinite-dimensional Riccati equations and numerical approximations. SIAM Journal on Control and Optimization, 21:95-139, 1983. [50] K. Glover, R.F. Curtain and J.R. Partington. Realisation and approximation of linear infinite-dimensional systems with error bounds. SIAM Journal on Control and Optimization, 26:863-898, 1988. [51] P. Grabowski. On the spectral-Lyapunov approach to parametric optimization of distributed parameter systems. IMA Journal of Mathematical Control and Information, 7:317-338, 1990. [52] M. Green, K. Glover, D. Limebeer and J. Doyle. A J-spectral factorization approach to W^ control. SIAM Journal on Control and Optimization, 28:1350-1371, 1990. [53] S. Hansen and G. Weiss. New results on the operator Carleson measure criterion. IMA Journal of Mathematical Control and Information, 14:3-32,1997. [54] F. Huang. Strong asymptotic stability of linear dynamical systems in Banach spaces. Journal of Differential Equations, 104:307-324, 1993. [55] K. Ito. Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces. In Distributed Parameter Systems, F. Kappcl, K. Kunisch and W. Schappacher, editors, pages 151-166, Springer-Verlag, New York, 1987. [56] K. Ito and G. Propst. Legendre-tau-Pade approximations to the one-dimensional wave equation with boundary oscillators. Numerical Functional Analysis and Optimization, 90:57-70, 1997. [57] S.M. Joshi. Control of Large Flexible Space Structures, Lecture Notes in Control and Information Science 131. Springer-Verlag, Berlin, 1989.
Bibliography
145
[58] F. Kappel and D. Salamon. An approximation theorem for the algebraic Riccati equation. SIAM Journal on Control and Optimization, 28:1136—1147, 1990. [59] T. Kato. Perturbation Theory of Linear Operators. Springer- Verlag, New York, 1966. [60] V. Kucera. New results in state estimation and regulation. Automatica, 17:745-748, 1981. [61] J.P. Lasalle. The Stability of Dynamical Systems. SIAM, Philadelphia, PA, 1976. [62] N. Levan. The stabilizability problem: A Hilbert space operator decomposition approach. IEEE Transactions on Circuits and Systems, 25:721-727, 1978. [63] N. Levan. On some relationships between the LaSalle invariance principle and the Nagy-Foias canonical decomposition. Journal of Mathematical Analysis and Applications, 77:493-504, 1980. [64] N. Levan. Approximate stabilizability via the algebraic Riccati equation. SIAM Journal on Control and Optimization, 23:153-160, 1985. [65] N. Levan. Strong stability of quasi-affine transforms of contraction semigroups and the steady-state Riccati equation. Journal of Optimization Theory and Applications, 45:397-406, 1985. [66] N. Levan. Stabilization by the feedbacks — B* and —B*P. In System Modelling and Optimization, Lecture Notes in Control and Information Science 84, Springer-Verlag, New York, pages 542-550. 1986. [67] N. Levan. Stability enhancement by state feedback. In Control Problems for Systems Described by Partial Differential Equations and Applications, pages 313-324. Springer-Verlag, Berlin, 1987. [68] N. Levan and L. Rigby. Strong stabilizability of linear contractive control systems on Hilbert space. SIAM Journal on Control and Optimization, 17:23-35, 1979. [69] J.L. Lions. Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin, 1971. [70] H. Logemann, E.P. Ryan and S. Townley. Integral control of infinite-dimensional linear systems subject to input saturation. SIAM Journal on Control and Optimization, 36:1940-1961, 1998. [71] J.M. Maciejowski. Multivariable Feedback Design. Addison-Wesley, Reading, MA, 1989. [72] D.C. McFarlane and K. Glover. Robust Controller Design using Normalized Coprime Factor Plant Descriptions, Lecture Notes in Control and Information Sciences 138. Springer-Verlag, New York, 1989. [73] K. Mikkola. On the Stable H-2 and H^ Infinite-Dimensional Regular Problems and their Algebraic Riccati Equations. Technical Research Report A383, Helsinki University of Technology, Institute of Mathematics, Helsinki, Finland, 1997. [74] B.P. Molinari. Equivalence relations for the algebraic Riccati equation. SIAM Journal on Control, 11:272-285, 1973.
146
Bibliography
[75] O. Morgiil. Dynamic boundary control of a Euler-Bernoulli beam. IEEE Transactions on Automatic Control, 37:639-642, 1992. [76] 6. Morgiil. Dynamic boundary control of the Timoshenko beam. Automatica, 28:1255-1260, 1992. [77] 6. Morgiil. A dynamic control law for the wave equation. Automatica, 30:1785-1792, 1994. [78] P.M. Morse and K.U. Ingard. Theoretical Acoustics. McGraw-Hill, New York, 1968. [79] K.S. Narendra and J.H. Taylor. Frequency Domain Criteria for Absolute Stability. Academic Press, New York, 1973. [80] J.C. Oostveen and R.F. Curtain. Riccati equations for strongly stabilizable bounded linear systems. Automatica, 34:953-967, 1998. [81] J.C. Oostveen and R.F. Curtain. Robustly stabilizing controllers for dissipative infinite-dimensional systems with collocated actuators and sensors. Automatica, 36:337-348, 2000. [82] J.C. Oostveen and R.F. Curtain. The Popov criterion for strongly stable distributed parameter systems. InternationalJournal of Control. Submitted for publication. [83] J.C. Oostveen, R.F. Curtain and K. Ito. An approximation theory for strongly stabilizing solutions to the operator LQ Riccati equation. SIAM Journal on Control and Optimization, to appear. [84] J.R. Partington. An Introduction to Hankel Operators. London Mathematical Society Student Texts. Cambridge University Press, Cambridge, UK, 1988. [85] A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1983. [86] V.M. Popov. Absolute stability of nonlinear systems of automatic control. Automation and Remote Control, 22:857-875, 1961. [87] S.C. Power. Hankel Operators on Hilbert Space. Pitman, Boston, MA, 1982. [88] A. Ran. Hankel norm approximation for infinite-dimensional systems and WienerHopf factorization. In Modelling Robustness and Sensitivity Reduction in Control Systems, R.F. Curtain, editor, NATO ASI Series, pages 57-70. Springer-Verlag, New York, 1986. [89] R. Rebarber. Conditions for the equivalence of internal and external stability for distributed parameter systems. IEEE Transactions on Automatic Control, 38:994998, 1993. [90] M. Rosenblum and J. Rovnyak. Hardy Classes and Operator Theory. Oxford University Press, New York, 1985. [91] D.L. Russell. Linear stabilization of the linear oscillator in Hilbert space. Journal of Mathematical Analysis and Applications, 25:663-675, 1969.
Bibliography
147
[92] M. Slemrod. The linear stabilization problem in Hilbert space. Journal of Functional Analysis, 11:334-345,1972. [93] M. Slemrod. A note on complete controllability and stabilizability for linear control systems in Hilbert space. SI AM Journal on Control, 12:500-508, 1974. [94] M. Slemrod. Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control. Mathematics of Control, Signals and Systems, 2:265-285, 1989. [95] M.C. Smith. On stabilization and the existence of coprime factorizations. IEEE Transactions on Automatic Control, 34:1005-1007, 1989. [96] OJ. Staffans. Quadratic optimal control of stable systems through spectral factorization. Mathematics of Control, Signals and Systems, 8:167-197, 1995. [97] O.J. Staffans. Quadratic optimal control of stable well-posed linear systems. Transactions of the American Mathematical Society, 349:3679-3716, 1997. [98] O.J. Staffans. Coprime factorizations and well-posed linear systems. SIAM Journal on Control and Optimization, 36:1268-1292, 1998. [99] O.J. Staffans. Feedback representations of critical controls for well-posed linear sy stems. International Journal of Robust an d Nonlinear Control, 8:1189-1217,1998. [100] O.J. Staffans. On the distributed stable full information H^ problem. International Journal of Robust and Nonlinear Control, 8:1255-1305, 1998. [101] O.J. Staffans. Quadratic optimal control of well-posed linear systems. SIAM Journal on Control and Optimization, 37:131-164, 1998. [102] S.R. Treil. The theorem of Adamjan-Arov-Krein: A vector variant. Zap. Nauchn. Semin. Leningrad. Otdel. Math. Inst. Steklov (LOMI), 141:56-71, 1985 (in Russian). [103] B. van Keulen. Hankel operators for non-exponentially stabilizable infinitedimensional systems. Systems and Control Letters, 15:221-226, 1990. [104] B. van Keulen. H^-Control for Distributed Parameter Systems: A State-space Approach. Birkhauser, Boston, MA, 1993. [105] M. Vidyasagar. Nonlinear Systems Analysis, second edition. Prentice-Hall, Englewood Cliffs, NJ, 1993. [106] G. Weiss. Admissibility of unbounded control operators. SIAM Journal on Control and Optimization, 27:527-545, 1989. [107] G. Weiss. Admissible observation operators for linear semigroups. Israel Journal of Mathematics, 65:17^3, 1989. [108] G. Weiss. The representation of regular linear systems on Hilbert spaces. In Control and Estimation of Distributed Parameter Systems, W. Schappacher, F. Kappel and K. Kunisch, editors, pages 401-416. Birkhauser-Verlag, Basel, Switzerland, 1989.
148
Bibliography
[109] G. Weiss. Representation of shift-invariant operators on Z/2 by H^ transfer functions: An elementary proof, a generalization to Lp and a counterexample for LOOMathematics of Control, Signals and Systems, 4:193-203, 1991. [110] G. Weiss. Regular linear systems with feedback. Mathematics of Control, Signals and Systems, 7:23-57, 1994. [Ill] M. Weiss. Riccati Equations in Hilbert Spaces: A Popov function approach. PhD thesis, Rijksuniversiteit Groningen, The Netherlands, 1994. [112] M. Weiss. Riccati equation theory for Pritchard-Salamon systems: A Popov function approach. IMA Journal of Mathematical Control and Information, 14:45-84, 1997. [113] G. Weiss and R.F. Curtain, Stabilizing controllers with internal loop for infinitedimensional systems. In Proceedings of the European Control Conference, pages 3294-3297, 1995. [114] M. Weiss and G. Weiss. Optimal control of stable weakly regular linear systems. Mathematics of Control, Signals and Systems, 10:287-330, 1997. [115] D. Wexler. Frequency domain stability for a class of equations arising in reactor dynamics. SIAM Journal on Mathematical Analysis, 10:118-138, 1979. [116] D. Wexler. On frequency domain stability for evolution equations in Hilbert spaces via the algebraic Riccati equation. SIAM Journal on Mathematical Analysis, 11:969— 983, 1980. [117] J.C. Willems. Dissipative dynamical systems. Part I: General theory. Archive for Rational Mechanics and Analysis, 45:321-351, 1972. [118] J.C. Willems. Dissipative dynamical systems. Part II: Linear systems with quadratic supply rates. Archive for Rational Mechanics and Analysis, 45:352-393, 1972. [119] J.L. Willems. Stability Theory of Dynamical Systems. Nelson, London, 1970. [120] Y. You. Dynamical boundary control of two-dimensional Petrovsky system: Vibrating rectangular plate. In Analysis and Optimization of Systems, A. Bensoussan and J.L. Lions, editors, Lecture Notes in Control and Information Sciences 111, pages 519-530. Springer-Verlag, Heidelberg, 1988.
Index acoustical problem, 120 admissible feedback transfer function, 68 approximation of strongly stabilizing solution, 60 of strongly stable system, 55 approximation of Riccati equation, 51 asymptotic tracking, 112
example flexible beam, 115 flexible plate, 117 numerical approximation, 123 physical, 115 propagation of sound, 120 exponential stability, 1 extended input-output map, 11 extended input map, 10 extended output map, 10
bounded real, 46 strictly, 46 bounded real lemma, 46
factorization coprime, 67 J-spectral, 44 Wiener-Hopf, 32 flexible beam, 115 flexible plate, 117 flexible structures, 4, 15, 115, 117, 129
central controller, 103 coercivity condition, 32 collocated actuators and sensors, 4, 15 compact resolvent, 17 compensator, 68 contraction semigroup, 17 controllability Gramian, 87 coprime factorization, 67 doubly, 68 left-, 67 normalized doubly, 68 normalized left-, 67 normalized right-, 68 right-, 67 state-space formulas, 69
HOC control, 139 Hankel operator frequency-domain, 87 time-domain, 88 hybrid system, 65, 129 input stability, 11 input-output stability, 11 input-output stabilization, 69 integral control, 112
detectable strongly, 14, 23 dissipation inequality, 4 dissipative systems with collocated actuators and sensors, 15,62 dissipative operator, 17 dissipativity, 16 doubly coprime factorization, 68 dual system, 36 dynamic compensator, 68 dynamic stabilization, 77
J-spectral factorization, 44 left-coprime factorization, 67 LQ Riccati equation, 47 Lyapunov equation, 86 Nehari extension problem, 86 Nehari problem, 85
149
150 normalized doubly coprime factorization, 68 left-coprime factorization, 67 right-coprime factorization, 68 numerical approximation of Riccati equation, 51 observability Gramian, 87 operator dissipative, 16, 17 output stability, 11 passivity, 4 perturbations coprime factor, 85 left-coprime factor, 96 nonlinear, 105 Popov criterion, 107 Popov function, 31 positive real strictly, 16 positive real lemma, 46, 49 propagation of sound, 120 regular linear system, 36 Riccati equation approximation, 123 approximation of solution, 51, 60 control, 72 filter, 72 LQ, 26,47,49,51, 123 stable case, 39 statically stabilizable case, 31,42 strongly stabilizing solution, 31 right-coprime factorization, 67 robustness coprime factor perturbations, 96 nonlinear perturbations, 105 robustness margin, 96 robust stabilization, 85 sectorial nonlinearity, 107 semigroup of contractions, 17 solution mild, 106 strong, 106
Index stability exponential, 1 global asymptotic, 107 input, 11 input-output, 11 in the large, 106 output, 11 strong stability of semigroup, 2, 11 strong stability of system, 11 uniform asymptotic in the large, 107 uniform input-output, 57 uniform output, 56 weak stability of semigroup, 3 stabilizable statically, 14, 23 strongly, 14, 23 stabilizable-detectable strongly, 54 stabilization dynamic, 77 input-output, 69 robust, 85, 96 statically stabilizable, 14, 23 strictly bounded real, 46 strictly positive real, 16 strongly detectable, 14, 23 strongly stabilizable, 14, 23 strongly stabilizable-detectable, 54 strongly stabilizing solution, 31 strongly stable semigroup, 11 strongly stable system, 11 strong convergence of systems, 56 strong stability of semigroup, 2 Toeplitz operator time-domain, 33 uniform input-output stability, 57 uniform output stability, 56 weak stability, 138 weak stability of semigroup, 3 well-posed linear system, 35 Wiener-Hopf factorization, 32 Youla parameterization, 69 zeros, 22